Recent zbMATH articles in MSC 57https://www.zbmath.org/atom/cc/572021-11-25T18:46:10.358925ZWerkzeugA singular mathematical promenadehttps://www.zbmath.org/1472.000012021-11-25T18:46:10.358925Z"Ghys, Étienne"https://www.zbmath.org/authors/?q=ai:ghys.etienneAt the first look, one may feel that the book title is a little bit strange. The word singular in the title refers to the concept of singularity of a curve and does not mean a trip made by an individual person. It is a promenade into the mathematical world. The tour is interesting, entertaining and enjoyable, but it may be little bit difficult for those who have insufficient mathematical knowledge. So some mathematical maturity is required to fully appreciate the beauty presented by the author. When you go through the subjects of it you will find it a wonderfully crafted book. The book consists of 30 chapters. Each chapter provides a rich read. Several chapters are fairly independent from the rest of the book. It is a remarkable achievement in terms of its content, structure, and style. In almost all chapters the author shows excellent examples of mathematical exposition and utilize history to enrich a contemporary mathematical investigations. Actually he weaves historical stories in between the combinatorics, complex analysis, and algebraic geometry \dots etc. and does it all in a very readable and remarkable way. The design of the book is amazing: it contains many pictures and illustrations, scanned manuscripts, references, remarks, all written in the right margin of the pages (so one has the information immediately available). The text contains many historical quotations in different languages, with translations, and interesting analysis of the mathematics of our ``classics'' (Newton, Gauss, Hipparchus \dots etc). Hence the book will please any budding or professional mathematician. I can say that, principally, for professional readers, the book is an enjoyable reading due to the versatility of subjects using too many illustrations and remarks that enriched the concepts of the classical notions. In fact most of the material in the book can be regarded as an advanced undergraduate/early graduate level, even there are some material that is significantly more advanced. One very remarkable aspects of the book is the treat of historical matters. Some of the very classical notions such as the fundamental theorem of algebra, the theory of Puisseux series, the linking number of knots, discrete mathematics, operads, resolution of curve singularities, complex singularities, and more, have been discussed and explained in an enlightening way.
The author of the book, Professor Étienne Ghys, Director of Research at the École Normale Superiere de Lyon, is a skilled, gifted versatile expositor mathematician. He wrote his book in a relaxed, informal manner with lots of exclamation marks, figures, supporting computer graphics and illustrations that are mathematically helpful and visually engaging. It is interesting to know that most of illustrations have been produced by Ghys himself and who has waived all copyright and related or neighboring rights which is a good evidence of Ghys's service towards the dissemination of mathematical ideas. Ghys is a prominent researcher, broadly in geometry and dynamics. He was awarded the Clay Award for Dissemination of Mathematics in 2015.
As the author mentioned in his book, the motivation for writing such an interesting book came from a fact brought to his attention by his colleague, Maxim Kontsevich, in 2009 that relates the relative position of the graphs of four real polynomials under certain conditions imposed on the polynomials. So he begins the book with an attractive theorem of Maxim Kontsevich scribbled for him on a Paris metro ticket who stated it in a nice: Theorem. There do not exist four polynomials \(P_1, \dots , P_4 \in R[x] \) with \(P_1(x) < P_2(x) < P_3(x) < P_4(x)\) for all small negative \(x\), and \(P_2(x) < P_4(x) < P_1(x) < P_3(x)\) for small positive \(x\).
In fact Ghys begins his promenade with this attractive theorem. Amazingly, this result basically characterizes what can or cannot happen for crossings, not only for graphs of arbitrary collections of polynomials, but indeed for all real analytic planar curves. Actually the book explores very different questions related to this problem, and follows on different ramifications. Ghys discussed the more general singularities of algebraic curves in the plane, explaining how the concepts were developed historically. I recommend to assign parts of it as an independent studies for both undergraduate and graduate students.Existential graphs on nonplanar surfaceshttps://www.zbmath.org/1472.030092021-11-25T18:46:10.358925Z"Oostra, Arnold"https://www.zbmath.org/authors/?q=ai:oostra.arnoldSummary: Existential graphs on the plane constitute a two-dimensional representation of classical logic, in which a Jordan curve stands for the negation of its inside. In this paper we propose a program to develop existential Alpha graphs, which correspond to propositional logic, on various surfaces. The geometry of each manifold determines the possible Jordan curves on it, leading to diverse interpretations of negation. This may open a way for appointing a ``natural'' logic to any surface.Independence polynomials and Alexander-Conway polynomials of plumbing linkshttps://www.zbmath.org/1472.050792021-11-25T18:46:10.358925Z"Stoimenow, A."https://www.zbmath.org/authors/?q=ai:stoimenow.alexanderSummary: We use the Chudnovsky-Seymour Real Root Theorem for independence polynomials to obtain some statements about the coefficients and roots of the Alexander and Conway polynomial of some types of plumbing links, addressing conjectures of Fox, Hoste and Liechti.On null 3-hypergraphshttps://www.zbmath.org/1472.051292021-11-25T18:46:10.358925Z"Frosini, Andrea"https://www.zbmath.org/authors/?q=ai:frosini.andrea"Kocay, William L."https://www.zbmath.org/authors/?q=ai:kocay.william-l"Palma, Giulia"https://www.zbmath.org/authors/?q=ai:palma.giulia"Tarsissi, Lama"https://www.zbmath.org/authors/?q=ai:tarsissi.lamaSummary: Given a 3-uniform hypergraph \(H\) consisting of a set \(V\) of vertices, and \(T \subseteq \binom{V}{3}\) triples, a null labelling is an assignment of \(\pm 1\) to the triples such that each vertex is contained in an equal number of triples labelled \(+ 1\) and \(- 1\). Thus, the signed degree of each vertex is zero. A necessary condition for a null labelling is that the degree of every vertex of \(H\) is even. The Null Labelling Problem is to determine whether \(H\) has a null labelling. It is proved that this problem is NP-complete. Computer enumerations suggest that most hypergraphs which satisfy the necessary condition do have a null labelling. Some constructions are given which produce hypergraphs satisfying the necessary condition, but which do not have a null labelling. A self complementary 3-hypergraph with this property is also constructed.An extension of Stanley's chromatic symmetric function to binary delta-matroidshttps://www.zbmath.org/1472.051482021-11-25T18:46:10.358925Z"Nenasheva, M."https://www.zbmath.org/authors/?q=ai:nenasheva.m"Zhukov, V."https://www.zbmath.org/authors/?q=ai:zhukov.v-a.1|zhukov.v-o|zhukov.vadim-g|zhukov.vladimir-v|zhukov.v-d|zhukov.vitalii-vladimirovich|zhukov.v-e|zhukov.v-t|zhukov.victor-p|zhukov.vyacheslav|zhukov.v-iSummary: Stanley's symmetrized chromatic polynomial is a generalization of the ordinary chromatic polynomial to a graph invariant with values in a ring of polynomials in infinitely many variables. The ordinary chromatic polynomial is a specialization of Stanley's one.
To each orientable embedded graph with a single vertex, a simple graph is associated, which is called the intersection graph of the embedded graph. As a result, we can define Stanley's symmetrized chromatic polynomial for any orientable embedded graph with a single vertex. Our goal is to extend Stanley's chromatic polynomial to embedded graphs with arbitrary number of vertices, and not necessarily orientable. In contrast to well-known extensions of, say, the Tutte polynomial from abstract to embedded graphs [\textit{C. Chun} et al., J. Comb. Theory, Ser. A 167, 7--59 (2019; Zbl 1417.05103)], our extension is based not on the structure of the underlying abstract graph and the additional information about the embedding. Instead, we consider the binary delta-matroid associated to an embedded graph and define the extended Stanley chromatic polynomial as an invariant of binary delta-matroids. We show that, similarly to Stanley's symmetrized chromatic polynomial of graphs, which satisfies 4-term relations for simple graphs, the polynomial that we introduce satisfies the 4-term relations for binary delta-matroids [\textit{S. Lando} and \textit{V. Zhukov}, Mosc. Math. J. 17, No. 4, 741--755 (2017; Zbl 1414.05067)].
For graphs, Stanley's chromatic function produces a knot invariant by means of the correspondence between simple graphs and knots. Analogously we may interpret the suggested extension as an invariant of links, using the correspondence between binary delta-matroids and links.Differentiable approximation of continuous semialgebraic mapshttps://www.zbmath.org/1472.140672021-11-25T18:46:10.358925Z"Fernando, José F."https://www.zbmath.org/authors/?q=ai:fernando.jose-f"Ghiloni, Riccardo"https://www.zbmath.org/authors/?q=ai:ghiloni.riccardoThis paper concerns the problem of approximating a uniformly continuous semialgebraic map \(f: S \to T\) from a compact semialgebraic set \(S\) to an arbitrary semialgebraic set \(T\) by a semialgebraic map \(g: S \to T\) that is differentiable of class \(C^\nu\), where \(\nu\) is a positive integer or \(\infty\). It is known that if \(T\) is an \(C^\nu\) semialgebraic manifold, then arbitrarily good (in the \(C^\nu\)-norm) \(C^\nu\) semialgebraic approximations exist. The authors show that for \emph{any semialgebraic \(T\)}, arbitrarily good \(\nu = 1\) approximations are possible. For \(\nu \geq 2\), they obtain density results when: (1) \(T\) is compact and locally \(C^\nu\) semialgebraically equivalent to a polyhedron, or (2) \(T\) is an open semialgebraic subset of a Nash set. The paper includes a useful review of approximation results in semialgebraic geometry, including discussion of key references.\(\mathfrak{sl}_N\)-web categories and categorified skew Howe dualityhttps://www.zbmath.org/1472.170582021-11-25T18:46:10.358925Z"Mackaay, Marco"https://www.zbmath.org/authors/?q=ai:mackaay.marco"Yonezawa, Yasuyoshi"https://www.zbmath.org/authors/?q=ai:yonezawa.yasuyoshiSummary: In this paper we show how the colored Khovanov-Rozansky \(\mathfrak{sl}_N\)-matrix factorizations, due to \textit{Hao Wu} [Diss. Math. 499, 215 p. (2014; Zbl 1320.57020)] and \textit{Y. Yonezawa} [Nagoya Math. J. 204, 69--123 (2011; Zbl 1271.57033)], can be used to categorify the type \(A\) quantum skew Howe duality defined by Cautis, Kamnitzer and Morrison in [\textit{S. Cautis} et al., Math. Ann. 360, No. 1--2, 351--390 (2014; Zbl 1387.17027)]. In particular, we define \(\mathfrak{sl}_N\)-web categories and 2-representations of Khovanov and Lauda's categorical quantum \(\mathfrak{sl}_m\) on them. We also show that this implies that each such web category is equivalent to the category of finite-dimensional graded projective modules over a certain type \(A\) cyclotomic KLR-algebra.Ping-pong configurations and circular orders on free groupshttps://www.zbmath.org/1472.200572021-11-25T18:46:10.358925Z"Malicet, Dominique"https://www.zbmath.org/authors/?q=ai:malicet.dominique"Mann, Kathryn"https://www.zbmath.org/authors/?q=ai:mann.kathryn"Rivas, Cristóbal"https://www.zbmath.org/authors/?q=ai:rivas.cristobal"Triestino, Michele"https://www.zbmath.org/authors/?q=ai:triestino.micheleSummary: We discuss actions of free groups on the circle with ``ping-pong'' dynamics; these are dynamics determined by a finite amount of combinatorial data, analogous to Schottky domains or Markov partitions. Using this, we show that the free group \(F_n\) admits an isolated circular order if and only if \(n\) is even, in stark contrast with the case for linear orders. This answers a question from [\textit{K. Mann} and \textit{C. Rivas}, Ann. Inst. Fourier 68, No. 4, 1399--1445 (2018; Zbl 07002300)]. Inspired by work in [\textit{S. Alvarez} et al., ``Generalized ping-pong partitions and locally discrete groups of real-analytic circle diffeomorphisms. II: Applications'', Preprint, \url{arXiv:2104.03348}], we also exhibit examples of ``exotic'' isolated points in the space of all circular orders on \(F_2\). Analogous results are obtained for linear orders on the groups \(F_n \times \mathbb{Z}\).Applications of the Alexander ideals to the isomorphism problem for families of groupshttps://www.zbmath.org/1472.200682021-11-25T18:46:10.358925Z"Hung, Do Viet"https://www.zbmath.org/authors/?q=ai:hung.do-viet"Khoi, Vu The"https://www.zbmath.org/authors/?q=ai:khoi.vu-theSummary: In this paper we use the Alexander ideals of groups to solve the isomorphism problem for the Baumslag-Solitar groups and a family of parafree groups introduced by \textit{G. Baumslag} and \textit{S. Cleary} [J. Group Theory 9, No. 2, 191--201 (2006; Zbl 1109.20025)].The infinite Fibonacci groups and relative asphericityhttps://www.zbmath.org/1472.200702021-11-25T18:46:10.358925Z"Edjvet, Martin"https://www.zbmath.org/authors/?q=ai:edjvet.martin"Juhász, Arye"https://www.zbmath.org/authors/?q=ai:juhasz.aryeSummary: We prove that the generalised Fibonacci group \(F(r,n)\) is infinite for \((r,n)\in\{(7+5k,5),(8+5k,5):k\geq0\}\). This together with previously known results yields a complete classification of the finite \(F(r,n)\), a problem that has its origins in a question by J. H. Conway in 1965. The method is to show that a related relative presentation is aspherical from which it can be deduced that the groups are infinite.Higher finiteness properties of braided groupshttps://www.zbmath.org/1472.200742021-11-25T18:46:10.358925Z"Bux, Kai-Uwe"https://www.zbmath.org/authors/?q=ai:bux.kai-uweSummary: The properties of a group to be finitely generated or finitely presentable are the first two instances in a sequence of so called \textit{higher finiteness properties} defined in terms of skeletons of classifying spaces. The study of higher finiteness properties is a prime example of how one can use a nice action of a group on a topological space to better understand the group. We shall illustrate this method in detail using the braided Thompson group \(V^{\text{br}}\) as an example.
For the entire collection see [Zbl 1416.60012].On laminar groups, Tits alternatives and convergence group actions on \(\mathrm{S}^2\)https://www.zbmath.org/1472.200792021-11-25T18:46:10.358925Z"Alonso, Juan"https://www.zbmath.org/authors/?q=ai:alonso.juan-i|alonso.juan-miguel|alonso.juan-antonio|alonso.juan-j|peral-alonso.juan-carlos"Baik, Hyungryul"https://www.zbmath.org/authors/?q=ai:baik.hyungryul"Samperton, Eric"https://www.zbmath.org/authors/?q=ai:samperton.ericSummary: Following previous work of the second author, we establish more properties of groups of circle homeomorphisms which admit invariant laminations. In this paper, we focus on a certain type of such groups, so-called pseudo-fibered groups, and show that many 3-manifold groups are examples of pseudo-fibered groups. We then prove that torsion-free pseudo-fibered groups satisfy a Tits alternative. We conclude by proving that a purely hyperbolic pseudo-fibered group acts on the 2-sphere as a convergence group. This leads to an interesting question if there are examples of pseudo-fibered groups other than 3-manifold groups.Erratum to: ``The braided Thompson's groups are of type \(F_\infty \)''https://www.zbmath.org/1472.200822021-11-25T18:46:10.358925Z"Bux, Kai-Uwe"https://www.zbmath.org/authors/?q=ai:bux.kai-uwe"Witzel, Stefan"https://www.zbmath.org/authors/?q=ai:witzel.stefan"Zaremsky, Matthew C. B."https://www.zbmath.org/authors/?q=ai:zaremsky.matthew-curtis-burkholderFrom the text: There is a mistake in Lemma 3.9 of the authors' paper [ibid. 718, 59--101 (2016; Zbl 1397.20053)], which has no consequences for the rest of the article. The assumption that the link of every \(k\)-simplex in \(X\) be \((m-2k-2)\)-connected is insufficient to get the induction step to work, and needs to be replaced by \((m-k-2)\)-connected.The basilica Thompson group is not finitely presentedhttps://www.zbmath.org/1472.200862021-11-25T18:46:10.358925Z"Witzel, Stefan"https://www.zbmath.org/authors/?q=ai:witzel.stefan"Zaremsky, Matthew C. B."https://www.zbmath.org/authors/?q=ai:zaremsky.matthew-curtis-burkholderSummary: We show that the Basilica Thompson group introduced by \textit{J. Belk} and \textit{B. Forrest} [Groups Geom. Dyn. 9, No. 4, 975--1000 (2015; Zbl 1366.20026)] is not finitely presented, and in fact is not of type \(\mathrm{FP}_2\). The proof involves developing techniques for proving non-simple connectedness of certain subcomplexes of CAT(0) cube complexes.A user's guide to cloning systemshttps://www.zbmath.org/1472.200872021-11-25T18:46:10.358925Z"Zaremsky, Matthew C. B."https://www.zbmath.org/authors/?q=ai:zaremsky.matthew-curtis-burkholderSummary: The author, in joint work with \textit{S. Witzel} [Groups Geom. Dyn. 12, No. 1, 289--358 (2018; Zbl 1456.20050)], developed a procedure for building new examples of groups in the extended family of Thompson groups, using what we termed cloning systems. These new Thompson-like groups can be thought of as limits of families of groups; however, unlike other limiting processes, e.g., direct limits, these tend to be well behaved with respect to finiteness properties. In this expository paper, we distill the crucial parts of that 50-page paper into a more digestible form for those curious to understand the construction but less curious about the gritty details. We also give some new examples involving signed symmetric groups and twisted braid groups.Acylindrical actions on projection complexeshttps://www.zbmath.org/1472.200892021-11-25T18:46:10.358925Z"Bestvina, Mladen"https://www.zbmath.org/authors/?q=ai:bestvina.mladen"Bromberg, Kenneth"https://www.zbmath.org/authors/?q=ai:bromberg.kenneth-w"Fujiwara, Koji"https://www.zbmath.org/authors/?q=ai:fujiwara.koji.1"Sisto, Alessandro"https://www.zbmath.org/authors/?q=ai:sisto.alessandroSummary: We simplify the construction of projection complexes from [\textit{M. Bestvina} et al., Publ. Math., Inst. Hautes Étud. Sci. 122, 1--64 (2015; Zbl 1372.20029)]. To do so, we introduce a sharper version of the Behrstock inequality, and show that it can always be enforced. Furthermore, we use the new setup to prove acylindricity results for the action on the projection complexes.
We also treat quasi-trees of metric spaces associated to projection complexes, and prove an acylindricity criterion in that context as well.Hyperbolicities in CAT(0) cube complexeshttps://www.zbmath.org/1472.200902021-11-25T18:46:10.358925Z"Genevois, Anthony"https://www.zbmath.org/authors/?q=ai:genevois.anthonySummary: This paper is a survey dedicated to the following question: given a group acting on a CAT(0) cube complex, how to exploit this action to determine whether or not the group is Gromov/relatively/acylindrically hyperbolic? As much as possible, the different criteria we mention are illustrated by applications. We also propose a model for universal acylindrical actions of cubulable groups, and give a few applications to Morse, stable and hyperbolically embedded subgroups.Nonhyperbolic Coxeter groups with Menger boundaryhttps://www.zbmath.org/1472.200912021-11-25T18:46:10.358925Z"Haulmark, Matthew"https://www.zbmath.org/authors/?q=ai:haulmark.matthew"Hruska, G. Christopher"https://www.zbmath.org/authors/?q=ai:hruska.g-christopher"Sathaye, Bakul"https://www.zbmath.org/authors/?q=ai:sathaye.bakulSummary: A generic finite presentation defines a word hyperbolic group whose boundary is homeomorphic to the Menger curve. In this article we produce the first known examples of non-hyperbolic CAT(0) groups whose visual boundary is homeomorphic to the Menger curve. The examples in question are the Coxeter groups whose nerve is a complete graph on \(n\) vertices for \(n \geq 5\). The construction depends on a slight extension of Sierpinski's theorem on embedding 1-dimensional planar compacta into the Sierpinski carpet. We give a simplified proof of this theorem using the Baire category theorem.Growth of quasiconvex subgroupshttps://www.zbmath.org/1472.200922021-11-25T18:46:10.358925Z"Dahmani, François"https://www.zbmath.org/authors/?q=ai:dahmani.francois"Futer, David"https://www.zbmath.org/authors/?q=ai:futer.david"Wise, Daniel T."https://www.zbmath.org/authors/?q=ai:wise.daniel-tSummary: We prove that non-elementary hyperbolic groups grow exponentially more quickly than their infinite index quasiconvex subgroups. The proof uses the classical tools of automatic structures and Perron-Frobenius theory.
We also extend the main result to relatively hyperbolic groups and cubulated groups. These extensions use the notion of growth tightness and the work of the first author et al. [Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces. Providence, RI: American Mathematical Society (AMS) (2016; Zbl 1396.20041)] on rotating families.Polar actions on Damek-Ricci spaceshttps://www.zbmath.org/1472.220062021-11-25T18:46:10.358925Z"Kollross, Andreas"https://www.zbmath.org/authors/?q=ai:kollross.andreasThe author studies polar actions on Damek-Ricci spaces. A Damek-Ricci space is a non-compact harmonic Riemannian manifold. An isometric Lie group action on a Riemannian manifold is called polar if there exists a closed connected submanifold which meets all orbits orthogonally.
A Damek-Ricci space can be seen as a semi-direct product \(S=A\ltimes N\), where \(A\) is one dimensional and \(N\) is a generalized Heisenberg group. The Lie algebra \(\mathfrak{n}\) of \(N\) decomposes as \(\mathfrak{n}=\mathfrak{v}\oplus \mathfrak{z}\), where \(\mathfrak{v}\) is a vector space and \(\mathfrak{z}\) is the Lie algebra of the center of \(N\). The group \(N\) is equipped with an inner product, and, for \(Z\in \mathfrak{z}\), the endomorphism \(J_Z\) defined by \[\langle J_ZU,V\rangle =\langle [U,V],Z\rangle \quad (U,V\in \mathfrak{v}),\] satisfies \[J_Z^2=-\langle Z,Z\rangle Id_{\mathfrak{v}}.\] The author considers a closed connected subgroup \(\Sigma\) of \(S\) whose Lie algebra decomposes as \(\mathfrak{a}\oplus \mathfrak{v}'\oplus \mathfrak{z}'\), with \(\mathfrak{v}'\subset \mathfrak{v}\), \(\mathfrak{z}'\subset \mathfrak{z}\). Among other results it is shown that \(\Sigma \) is a totally geodesic submanifold of \(S\) if \(J_{\mathfrak{v}'}=\mathfrak{v}'\). Furthermore, if \(\mathfrak{z}'=0\), there exists a closed subgroup \(H\) of \(S\) which acts polarly on \(S\) by left translations such that \(\Sigma \) is a section for the action of \(H\).Spinorial representations of orthogonal groupshttps://www.zbmath.org/1472.220072021-11-25T18:46:10.358925Z"Ganguly, Jyotirmoy"https://www.zbmath.org/authors/?q=ai:ganguly.jyotirmoy"Joshi, Rohit"https://www.zbmath.org/authors/?q=ai:joshi.rohitLet \(G = G_0 \rtimes C_2\), with \(G_0\) a connected, simple, compact Lie group and \(C_2\) is the cyclic group of order 2. The authors, give criteria for whether an orthogonal irreducible representation \((\pi,V) \) of \(G\) lifts to \(Pin(V )\) in terms of the highest weights of \(\pi\), the quotient of the weight lattice by the root lattice and of the value of some character. From these criteria they compute the first and second Stiefel-Whitney classes of the representations of the orthogonal groups. They notice that a irreducible representation of a noncompact semisimple Lie group is spinorial if and only if its restriction to a maximal compact subgroup is spinorial. They also consider the case of non irreducible representations.The mapping class group action on \(\mathrm{SU}(3)\)-character varietieshttps://www.zbmath.org/1472.220132021-11-25T18:46:10.358925Z"Goldman, William M."https://www.zbmath.org/authors/?q=ai:goldman.william-m"Lawton, Sean"https://www.zbmath.org/authors/?q=ai:lawton.sean"Xia, Eugene Z."https://www.zbmath.org/authors/?q=ai:xia.eugene-zLet \(\Sigma\) be a compact oriented surface of genus \(g\) with boundary \(\partial \Sigma\), which has \(n \ge 1\) components. Let us consider the group of orientation-preserving homeomorphisms of \(\Sigma\) which fix \(\partial \Sigma\) pointwise. The mapping class group \(\Gamma\) of \(\Sigma\) is the group of connected components of this group of homeomorphisms. The authors also define for algebraic group \(G\) the relative character variety \(\mathcal M_{\mathcal C}(G)\).
W. Goldman conjectures many years ago that \(\Gamma\) acts ergodically on \(\mathcal M_{\mathcal C}(K)\) for compact Lie groups \(K\) and proved this conjecture for some compact Lie groups. Later some other results in this direction was proved. In this article the case \(K=\mathrm{SU}(3)\), \(g=1\) and \(n=1\) is considered. It is proved that aforementioned action of \(\Gamma\) in this case is ergodic with respect to the natural symplectic measure on the character variety.Maximally stretched laminations on geometrically finite hyperbolic manifoldshttps://www.zbmath.org/1472.300172021-11-25T18:46:10.358925Z"Guéritaud, François"https://www.zbmath.org/authors/?q=ai:gueritaud.francois"Kassel, Fanny"https://www.zbmath.org/authors/?q=ai:kassel.fannySummary: Let \(\Gamma_0\) be a discrete group. For a pair \((j,\rho)\) of representations of \(\Gamma_0\) into \(\operatorname{PO}(n,1)=\operatorname{Isom}(\mathbb{H}^n)\) with \(j\) geometrically finite, we study the set of \((j,\rho)\)-equivariant Lipschitz maps from the real hyperbolic space \(\mathbb{H}^n\) to itself that have minimal Lipschitz constant. Our main result is the existence of a geodesic lamination that is ``maximally stretched'' by all such maps when the minimal constant is at least \(1\). As an application, we generalize two-dimensional results and constructions of Thurston and extend his asymmetric metric on Teichmüller space to a geometrically finite setting and to higher dimension. Another application is to actions of discrete subgroups \(\Gamma\) of \(\operatorname{PO}(n,1)\times\operatorname{PO}(n,1)\) on \(\operatorname{PO}(n,1)\) by right and left multiplication: we give a double properness criterion for such actions, and prove that for a large class of groups \(\Gamma\) the action remains properly discontinuous after any small deformation of \(\Gamma\) inside \(\operatorname{PO}(n,1)\times\operatorname{PO}(n,1)\).Pluripotential theory on Teichmüller space. I: Pluricomplex Green functionhttps://www.zbmath.org/1472.300182021-11-25T18:46:10.358925Z"Miyachi, Hideki"https://www.zbmath.org/authors/?q=ai:miyachi.hidekiIn the paper [Bull. Am. Math. Soc., New Ser. 27, No. 1, 143--147 (1992; Zbl 0766.30016)], \textit{S. L. Krushkal} announced the following result on the pluricomplex Green function on Teichmüller space:
Theorem. Let \(T_{g, m}\) be the Teichmüller space of Riemann surfaces of analytically finite-type \((g,m)\), and let \(d_T\) be the Teichmüller distance on \(T_{g, m}\). Then, the pluricomplex Green function \(g_{T_{g, m}} \) on \(T_{g, m}\) satisfies \[g_{T_{g, m}}(x, y)\log \tanh d_T(x, y)\] for \(x, y \in g_{T_{g, m}}\).
The author has a programm to investigate of the pluripotential theory on Teichmüller space. In this first one of a series of works, the author gave an alternative approach to the Krushkal formula of the pluricomplex Green function on Teichmüller space. In comparison with the original approach by Krushkal, the strategy is more direct here. He first shows that the Teichmüller space carries a natural stratified structure of real-analytic submanifolds defined from the structure of singularities of the initial differentials of the Teichmüller mappings from a given point. Then he gives a description of the Levi form of the pluricomplex Green function using the Thurston symplectic form via Dumas' symplectic structure on the space of holomorphic quadratic differentials [\textit{D. Dumas}, Acta Math. 215, No. 1, 55--126 (2015; Zbl 1334.57020)].Moduli space of meromorphic differentials with marked horizontal separatriceshttps://www.zbmath.org/1472.320052021-11-25T18:46:10.358925Z"Boissy, Corentin"https://www.zbmath.org/authors/?q=ai:boissy.corentinA (compact) translation surface is a pair \((X, \omega)\), where \(X\) is a (compact) Riemann surface and \(\omega\) is a holomorphic 1-form on the surface. Locally integrating the form defines a flat metric on the surface, with conical singularities. If one only asks the form \(\omega\) to be \emph{meromorphic}, obtaining a non-compact translation surface with infinite area (if the poles of \(\omega\) are not all simple).
The study of such objects is motivated by the fact that they appear naturally when dealing with compactifications of the moduli space of translation surfaces. More precisely, if a sequence \((X_n,\omega_n)\) converges to the boundary of the moduli space in the Deligne-Mumford compactification, then the thick components of \(X_n\), appropriately rescaled, converge to meromorphic differentials, see, e.g., [\textit{A. Eskin} et al., Publ. Math., Inst. Hautes Étud. Sci. 120, 207--333 (2014; Zbl 1305.32007)].
In this article, the author studies the topology of the moduli spaces of translation surfaces with poles equipped with an extra combinatorial data: the choice, for each singularity of (an equivalence class of) horizontal separatrix, denoted \(\mathcal{H}^{\text{hor}}\). Here, by horizontal separatrix we mean either an horizontal geodesic line ending (or beginning) at a conical singularity or an horizontal geodesic going to infinity in the flat metric if the singularity is a non simple pole.
The main result of the paper is a complete characterization of the connected components of \(\mathcal{H}^{\text{hor}}\). Similarly to the case of compact translation surfaces, proven in [\textit{M. Kontsevich} and \textit{A. Zorich}, Invent. Math. 153, No. 3, 631--678 (2003; Zbl 1087.32010)], in the general case of genus greater than 1 and underlining surfaces not belonging to the hyperelliptic component, there are at most 2 connected components, classified but an invariant \(\operatorname{Sp}\) which is a generalization of the classical \(\operatorname{Arf}\). In the hyperelliptic case the extra symmetry of the underlining surface yields more components. Finally, the genus-0 case is the most complicated one, depending also on the combinatorics of the singularities.
The main result is obtained by reducing the problem to the study of some cyclic group coming for the forgetful map from \(\mathcal{H}^{\text{hor}}\) to \(\mathcal{H}^{\text{sing}}\), which is the space of translation surfaces with poles in which singularities have given names. The latter space has the same connected components of the moduli space of translation with poles, which were classified by the author in [Comment. Math. Helv. 90, No. 2, 255--286 (2015; Zbl 1323.30060)].
Generalizing two local surgeries introduced in the article by Kontsevich and Zorich [loc. cit.], called \emph{breaking up a zero} and \emph{bubbling a handle}, so that they can be performed also on poles, one constructs some important elements in the cyclic groups coming from the covering. Using some topological analysis, one then proceeds to show that these elements always exist. If the above elements generate the whole cyclic group, then the corresponding moduli space is connected. If they generate a finite index subgroup, then the index gives the number of connected components. Depending on the genus and on whether or not we are in the hyperelliptic case, components are distinguished by some topological invariant, related to the classical \(\operatorname{Arf}\) invariant and to the parity of the spin structure.Topics on Teichmüller spaceshttps://www.zbmath.org/1472.320062021-11-25T18:46:10.358925Z"Seppälä, Mika"https://www.zbmath.org/authors/?q=ai:seppala.mika(no abstract)Recurrence on infinite cyclic coveringshttps://www.zbmath.org/1472.370182021-11-25T18:46:10.358925Z"Fathi, Albert"https://www.zbmath.org/authors/?q=ai:fathi.albertGiven a compact manifold \(M\), the author considers an infinite cyclic covering \(\pi :\tilde{M} \to M\) and a metric \(d\) on \(M\) which comes from a Riemannian metric \(g\) on \(M\) metric. Moreover a metric \(\tilde{d}\) on \(\tilde{M}\) is defined by \(\tilde{g}\), the lift of \(g\) by the covering map \(\pi :\tilde{M} \to M \) which is defined by \((n,\tilde{x})\to \tilde{x} + n\), where \(n\in \mathbb{Z}\) and \(\tilde{x} \in \tilde{M}\). The aim of this work is to study the chain recurrence properties of the lift of a homeomorphism to an infinite cyclic cover. The problem is related with the Poincaré-Birkhoff theorem.
The author assumes that there is a fixed continuous function \(\tilde{\varphi}:\tilde{M}\to \mathbb{R} \), which satisfies \(\tilde{\varphi}(\tilde{x}+n) =\tilde{\varphi}(\tilde{x})+n \), for all \(\tilde{x} \in \tilde{M}\) and \(n\in \mathbb{Z}\). It is also assumed that \(h:M\to M\) is a homeomorphism that lifts to a homeomorphism \(\tilde{h}:\tilde{M}\to \tilde{M}\) with \(\tilde{h}\) commuting with the deck transformation of the covering \(\pi :\tilde{M} \to M \), which means that \(\tilde{h} (\tilde{x}+n)=\tilde{h} (\tilde{x})+n\) for all \(\tilde{x}\in \tilde{M}\) and all \(n\in \mathbb{Z}\).
A relevant result is Corollary 1.6, which is proved in Section 6. Let \(\tilde{\mathcal{R}}(\tilde{h})\) be the chain-recurrent set of \(\tilde{h}\). It is proved that, under the above assumptions, the set \(\tilde{\mathcal{R}}(\tilde{h})\) is not empty if and only if there exists an \(h\)-invariant probability measure \(\mu\) on \(M\), with rotation number Rot\((\mu)=0\), whose support is contained in a unique chain-recurrent component of \(h\). The author constructs an example showing that Corollary 1.6 might not be true for a non-cyclic abelian Galois covering.
For the entire collection see [Zbl 1446.37002].Transitive partially hyperbolic diffeomorphisms with one-dimensional neutral centerhttps://www.zbmath.org/1472.370342021-11-25T18:46:10.358925Z"Bonatti, Christian"https://www.zbmath.org/authors/?q=ai:bonatti.christian"Zhang, Jinhua"https://www.zbmath.org/authors/?q=ai:zhang.jinhuaThe aim of this paper is to study partially hyperbolic diffeomorphism with some specific properties. A diffeomorphism of a manifold of class \(C^1\) is said to be partially hyperbolic when the tangent bundle decomposes as the direct sum of a uniformly expanding bundle, a uniformly contracting bundle, and a bundle with intermediate behavior, the center. Furthermore it is called topologically neutral along the center if its iterates map short paths tangent to the center subbundle to short paths, uniformly on \(n\). A diffeomorphism is called transitive if it has dense orbits.
The authors classify topologically (namely up to conjugation by a homeomorphism) diffeomorphisms of closed three-manifolds that are partially hyperbolic, transitive, and have one-dimensional topological neutral center. Namely, up to finite lifts and iterates, these are conjugate to either skew products over a linear Anosov diffeomorphism of the torus with a rotation of the circle, or the time 1-map of a transitive topological Anosov flow. In arbitrary dimension, the authors prove that there exists a continuous metric along the center foliation that is dynamically invariant. Then this result is used for the classification in dimension three.Lorenz attractors and the modular surfacehttps://www.zbmath.org/1472.370402021-11-25T18:46:10.358925Z"Bonatti, Christian"https://www.zbmath.org/authors/?q=ai:bonatti.christian"Pinsky, Tali"https://www.zbmath.org/authors/?q=ai:pinsky.taliGerbes in geometry, field theory, and quantisationhttps://www.zbmath.org/1472.530342021-11-25T18:46:10.358925Z"Bunk, Severin"https://www.zbmath.org/authors/?q=ai:bunk.severinSummary: This is a mostly self-contained survey article about bundle gerbes and some of their recent applications in geometry, field theory, and quantisation. We cover the definition of bundle gerbes with connection and their morphisms, and explain the classification of bundle gerbes with connection in terms of differential cohomology. We then survey how the surface holonomy of bundle gerbes combines with their transgression line bundles to yield a smooth bordism-type field theory. Finally, we exhibit the use of bundle gerbes in geometric quantisation of 2-plectic as well as 1- and 2-shifted symplectic forms. This generalises earlier applications of gerbes to the prequantisation of quasi-symplectic groupoids.On the escape rate of geodesic loops in an open manifold with nonnegative Ricci curvaturehttps://www.zbmath.org/1472.530462021-11-25T18:46:10.358925Z"Pan, Jiayin"https://www.zbmath.org/authors/?q=ai:pan.jiayinThe escape rate \(E(M,x)\) of a noncompact and complete Riemannian manifold \((M,x)\) with an infinite fundamental group \(\pi_1(M,x)\) measures how fast the minimizing geodesic loops representing elements of the fundamental group escape from bounded balls centered at \(x\). It is defined as follows: \[ E(M,x) = \limsup_{|\gamma| \rightarrow \infty} \frac{d_H(x,c_\gamma)}{|\gamma|}\] where \(\gamma \in \pi_1(M,x)\), \(c_\gamma\) is the shortest geodesic loop representing \(\gamma\), \(|\gamma|\) is the length of \(c_\gamma\), and \(d_H\) is Hausdorff distance.
In this paper it is shown that if \((M,x)\) is a complete open manifold with nonnegative Ricci curvature and \(E(M,x) =0\), then \(\pi_1(M,x)\) is virtually abelian, that is, it contains an abelian subgroup of finite index.
The author also notes that one can apply the Cheeger-Gromoll splitting theorem [\textit{J. Cheeger} and \textit{D. Gromoll}, Ann. Math. (2) 96, 413--443 (1972; Zbl 0246.53049)] to prove that the fundamental group of a complete open manifold with nonnegative Ricci curvature is virtually abelian if one assumes the stronger hypothesis that the minimizing geodesic loops representing elements of the fundamental group are contained in a bounded set. This proof is contained in an appendix.
A second appendix contains estimates of the escape rates of several manifolds.A spinorial analogue of the Brezis-Nirenberg theorem involving the critical Sobolev exponenthttps://www.zbmath.org/1472.530572021-11-25T18:46:10.358925Z"Bartsch, Thomas"https://www.zbmath.org/authors/?q=ai:bartsch.thomas.2|bartsch.thomas.1"Xu, Tian"https://www.zbmath.org/authors/?q=ai:xu.tianOn a compact spin manifold \((M^m,g)\), the authors study solutions of the nonlinear Dirac equation \(D\psi=\lambda\psi+f(|\psi|)\psi+|\psi|^{\frac{2}{m-1}}\psi\) where \(\lambda\in\mathbb{R}\) and \(f=o(s^{\frac{2}{m-1}})\) as \(s\to\infty\). Such an equation is called in the paper (NLD) and is the Euler-Lagrange equation associated to some functional \(\mathcal{L}_\lambda(\psi)\). The authors show in Theorem 2.1 that if \(f\) satisfies some conditions (called \(f_1\),\(f_2\) and \(f_3\) in the paper), then (NLD) has at least one energy solution \(\psi_\lambda\) for every \(\lambda>0\). Also the map \(\mathbb{R}^+\to \mathbb{R}^+; \lambda\to \mathcal{L}_\lambda(\psi_\lambda)\) is continuous and nonincreasing on each interval \([\lambda_k,\lambda_{k+1})\). On the other hand, if \(f\) satisfies some other conditions (called \(f_1\),\(f_4\) and \(f_5\) in the paper), then (NLD) has at least one energy solution \(\psi_\lambda\) for every \(\lambda\in \mathbb{R}\setminus \{\lambda_k:k\leq 0\}\). The map \(\mathbb{R}\setminus \{\lambda_k:k\leq 0\}\to \mathbb{R}^+; \lambda\to \mathcal{L}_\lambda(\psi_\lambda)\) is continuous and nonincreasing on each interval \([\lambda_{k-1},\lambda_{k})\), if \(k\geq 2\), respectively \((\lambda_{k-1},\lambda_{k})\), if \(k\leq 1\).\(\text{String}^c\) structures and modular invariantshttps://www.zbmath.org/1472.530592021-11-25T18:46:10.358925Z"Huang, Ruizhi"https://www.zbmath.org/authors/?q=ai:huang.ruizhi"Han, Fei"https://www.zbmath.org/authors/?q=ai:han.fei.1|han.fei"Duan, Haibao"https://www.zbmath.org/authors/?q=ai:duan.haibaoIn the paper the authors study algebraic topology concepts revolving around String\(^c\), strong String\(^c\)-structures, weak String\(^c\)-structures, relations between strong and weak String\(^c\) manifolds, modular invariants and group actions on String\(^c\) manifolds; a string structure is a higher version of a Spin structure related to quantum anomalies in physics. They look at string structures from the perspective of Whitehead tower and also of the free loop space \(LM\) (lifting the structure group of the looped spin frame bundle from the loop group of Spin\(^c(n)\) to its universal central extension). These two approaches of looking at string structures are equivalent when the manifold considered is 2-connected. The authors also extend the generalised Witten genera constructed for the first time by Chen.The dual Bonahon-Schläfli formulahttps://www.zbmath.org/1472.530862021-11-25T18:46:10.358925Z"Mazzoli, Filippo"https://www.zbmath.org/authors/?q=ai:mazzoli.filippoIn the paper, for a differentiable deformation of geometrically finite 3-manifold the Bonahon-Schläfli Formula gives the derivative of the volume of the convex cones in terms of the variation of the geometry of their boundaries (the classical Schläfli Formula is applicable for determining the volume of the hyperbolic polyhedra). Here the author studies the analogous problem for the dual volume and gives a self-contained proof of the dual Bonahon-Schläfli Formula (without using Bonahon's results).Asymptotic behavior of exotic Lagrangian tori \(T_{a,b,c}\) in \(\mathbb{C}P^2\) as \(a+b+c\to\infty\)https://www.zbmath.org/1472.530902021-11-25T18:46:10.358925Z"Lee, Weonmo"https://www.zbmath.org/authors/?q=ai:lee.weonmo"Oh, Yong-Geun"https://www.zbmath.org/authors/?q=ai:oh.yong-geun"Vianna, Renato"https://www.zbmath.org/authors/?q=ai:ferreira-de-velloso-vianna.renatoThe article under review is concerned with the asymptotic behavior of a certain family of monotone Lagrangian tori in \(\mathbb CP^2\). These Lagrangian tori are constructed in a previous work from [\textit{R. Ferreira de Velloso Vianna}, J. Topol. 9, No. 2, 535--551 (2016; Zbl 1350.53102)] and denoted by \(T_{a,b,c}\) where \((a,b,c)\) is a Markov triple satisfying \(a^2 + b^2 + c^2 = 3abc\). A deep result of \(T_{a,b,c}\) is that they are not pairwise Hamiltonian isotopic to each other. Therefore, it is a curious direction to investigate the geometric behavior of these \(T_{a,b,c}\) in \(\mathbb CP^2\) when \(a,b,c \to +\infty\). Here, \(\mathbb CP^2\) is associated with the standard Fubini-Study form \(\omega_{\mathrm{FS}}\) such that the area of the complex line is \(2\pi\). There are two main results in this article towards this direction.
Consider the following quantity as a symplectic invariant,
\[
c_G(\mathbb CP^2; T_{a,b,c}) : = \sup_{e}\{\pi r^2 \,| \, e: B^4(r) \to \mathbb CP^2 \backslash T_{a,b,c} \, \mbox{is a symplectic embedding}\},
\]
which is called the relative Gromov area of \(T_{a,b,c}\) in \(\mathbb CP^2\). Note that a similar but different symplectic invariant is studied by \textit{P. Biran} and \textit{O. Cornea} in [Geom. Topol. 13, No. 5, 2881--2989 (2009; Zbl 1180.53078)]. Then the first main result is that
\[
\inf_{a,b,c \to +\infty} c_G(\mathbb CP^2; T_{a,b,c}) \geq \frac{2\pi}{3}
\]
and it is conjectured that the above inequality is an equality. This result is not strong enough to conclude that the union of all \(T_{a,b,c}\) is not dense in \(\mathbb CP^2\), since for different \(T_{a,b,c}\) it might need different symplectic embeddings of \(B^4(r)\) to obtain the estimation above. However, the article indeed confirms that the union of certain Hamiltonian isotopic images of \(T_{a,b,c}\) is not dense in \(\mathbb CP^2\), via the geometric mutation theorem of the Lagrangian seeds in [\textit{V. Shende} et al., ``On the combinatorics of exact Lagrangian surfaces'', Preprint, \url{arXiv:1603.07449}].
The second main result in this article investigates the size of a Weinstein neighborhood of \(T_{a,b,c}\) in \(\mathbb CP^2\). In general, if \(L \subset M\) is a compact Lagrangian submanifold equipped with a metric \(g\), denote by \(\pi: T^*L \to L\) the canonical projection and define
\[
\mathfrak w_{\mathrm{DW}}(L; M) = \sup_{\Phi}\left\{ \inf_{q \in L} \left(\inf_{x \in \pi^{-1}(q) \cap \partial \mathcal U} \|\Phi(x)\|_{g(\pi(\Phi(x)))} \right)\right\}
\]
where the supremum is taken over all the Darboux-Weinstein charts \(\Phi: \mathcal U \to \mathcal V\) of \(L\) in \(M\). Then the result proves that
\[
\inf_{a,b,c \to +\infty} \mathfrak w_{\mathrm{DW}}(T_{a,b,c}; \mathbb CP^2) =0.
\]
Here, we view \(T_{a,b,c}\) as the image of Lagrangian embeddings from \(T^2\) with a fixed Riemannian metric. This illustrates a wild asymptotic behavior of \(T_{a,b,c}\), and its proof is based on the star-flux theory developed in [\textit{E. Shelukhin} et al., ``Geometry of symplectic flux and Lagrangian torus fibrations'', Preprint, \url{arXiv:1804.02044}]
In addition, this article also studies the ball packing problem in \(\mathbb CP^2 \backslash T_{a,b,c}\), which, by the definition of the symplectic blowup, is equivalent to the possibility of Lagrangian embeddings of \(T_{a,b,c}\) into \(\mathbb CP^2 \# k \overline{\mathbb CP^2}\) for some \(k \geq 0\). A result in this article shows that this is possible for any \(T_{a,b,c}\) if \(k \leq 5\). This answers a question from [\textit{Y. Chekanov} and \textit{F. Schlenk}, Electron. Res. Announc. Math. Sci. 17, 104--121 (2010; Zbl 1201.53083)].The quantitative nature of reduced Floer theoryhttps://www.zbmath.org/1472.530962021-11-25T18:46:10.358925Z"Venkatesh, Sara"https://www.zbmath.org/authors/?q=ai:venkatesh.saraSummary: We study the reduced symplectic cohomology of disk subbundles in negative symplectic line bundles. We show that this cohomology theory ``sees'' the spectrum of a quantum action on quantum cohomology. Precisely, quantum cohomology decomposes into generalized eigenspaces of the action of the first Chern class by quantum cup product. The reduced symplectic cohomology of a disk bundle of radius R sees all eigenspaces whose eigenvalues have size less than R, up to rescaling by a fixed constant. Similarly, we show that the reduced symplectic cohomology of an annulus subbundle between radii \(\operatorname{R}_1\) and \(\operatorname{R}_2\) captures all eigenspaces whose eigenvalues have size between \(\operatorname{R}_1\) and \(\operatorname{R}_2\), up to a rescaling. We show how local closed-string mirror symmetry statements follow from these computations.The space consisting of uniformly continuous functions on a metric measure space with the \(L^p\) normhttps://www.zbmath.org/1472.540062021-11-25T18:46:10.358925Z"Koshino, Katsuhisa"https://www.zbmath.org/authors/?q=ai:koshino.katsuhisaLet \(\mathbf{s} = (-1,1)^\mathbb{N}\) be a countable infinite product of lines endowed with the product topology and let \(c_0\) be the subspace of \(\mathbf{s}\) consisting of those sequences converging to zero. Kadec, Bessaga, Pelczynski and other well-known mathematicians studied homeomorphisms between various infinite dimensional Banach and Fréchet spaces motivated by several questions posed by Fréchet and Banach. A classical celebrated result due to \textit{R. D. Anderson} [Bull. Am. Math. Soc. 72, 515--519 (1966; Zbl 0137.09703)] and \textit{M. I. Kadets} [Funkts. Anal. Prilozh. 1, No. 1, 61--70 (1967; Zbl 0166.10603)] states that every separable infinite dimensional Banach space or Fréchet space is homeomorphic to \(\mathbf{s}\).
In contrast with this result, \textit{R. Cauty} showed in [Fundam. Math. 139, No. 1, 23--36 (1991; Zbl 0793.46008)] that the subspace of \(L^p[0,1]\) consisting of those elements having a representative which is continuous is homeomorphic to \(c_0\) for any \(1\leq p < \infty\). In this paper the author generalizes the aforementioned result of Cauty showing that if \(X\) is a metric measure space satisfying some natural conditions then the subspace \(C_u(X)\) of \(L^p(X)\) consisting of those elements having a representative which is uniformly continuous is homeomorphic to \(c_0\) for any \(1\leq p < \infty\).On the monotonicity of amenable categoryhttps://www.zbmath.org/1472.550022021-11-25T18:46:10.358925Z"Kotschick, D."https://www.zbmath.org/authors/?q=ai:kotschick.dieterThe notion of amenable category was introduced in [\textit{J. C. Gómez-Larrañaga} et al., Algebr. Geom. Topol. 13, No. 2, 905--925 (2013; Zbl 1348.55006)] as a variant of Lusternik-Schnirelmann category. It is defined via open covers of spaces for which the inclusions of the elements of the cover induce amenable subgroups on fundamental group level. In their recent article [``Amenable category and complexity'', Preprint, \url{arXiv:2012.00612}], \textit{P. Capovilla} et al. pose a question which is an analogue of Rudyak's monotonicity question for Lusternik-Schnirelmann category: Let \(M\) and \(N\) be closed manifolds. If there exists a map \(M \to N\) of non-zero degree, does it follow that \(\mathrm{cat}_{\mathrm{Am}}(M) \leq \mathrm{cat}_{\mathrm{Am}}(N)\)?
In the present article, the author gives a positive answer to this amenable monotonicity question in the three-dimensional case. To show this, he collects various results from [\textit{J. C. Gómez-Larrañaga} et al., loc. cit.]
to give explicit characterizations of 3-manifolds with \(\mathrm{cat}_{\mathrm{Am}}(M)=k\), where \(k \in \{1,2,3,4\}\). Combining these characterizations with known results about prime decompositions of 3-manifolds and known results about maps of non-zero degree, the author derives the claim straight from these computations.On the construction of a covering maphttps://www.zbmath.org/1472.550132021-11-25T18:46:10.358925Z"Saneblidze, Samson"https://www.zbmath.org/authors/?q=ai:saneblidze.samsonSummary: Let \(Y=\vert X\vert\) be the geometric realization of a path-connected simplicial set \(X\), and let \(G=\pi_{1}(X)\) be the fundamental group. Given a subgroup \(H\subset G\), let \(G/H\) be the set of cosets. Using the combinatorial model \(\boldsymbol{\Omega}X\to\mathbf{P}X\to X\) of the path fibration \({\Omega}Y\to{P}Y\to Y\) and a canonical action \(\mu\colon\boldsymbol{\Omega}X\times G/H\to G/H\), we construct a covering map \(G/H\to Y_{H}\to Y\) as the geometric realization of the associated short sequence \(G/H\to\mathbf{P}X\times_{\mu}G/H\to X\). This construction, in particular, does not use the existence of a maximal tree in \(X\). For a 2-dimensional \(X\) and \(H=\{1\}\), it can also be viewed as a simplicial approximation of a Cayley 2-complex of \(G\).A simplicial approach to stratified homotopy theoryhttps://www.zbmath.org/1472.550142021-11-25T18:46:10.358925Z"Douteau, Sylvain"https://www.zbmath.org/authors/?q=ai:douteau.sylvainThe author constructs a combinatorial model structure of stratified spaces. It uses the adjunction between stratified spaces and simplicial sets filtered over a poset. Applications are given, for instance to computations of filtered homotopy groups.Ribbon 2-knots, \(1+1=2\) and Duflo's theorem for arbitrary Lie algebrashttps://www.zbmath.org/1472.570012021-11-25T18:46:10.358925Z"Bar-Natan, Dror"https://www.zbmath.org/authors/?q=ai:bar-natan.dror"Dancso, Zsuzsanna"https://www.zbmath.org/authors/?q=ai:dancso.zsuzsanna"Scherich, Nancy"https://www.zbmath.org/authors/?q=ai:scherich.nancy-cLet \(\mathfrak{g}\) be a finite-dimensional Lie algebra. The Duflo isomorphism is an algebra isomorphism \(\mathcal{D}: S(\mathfrak{g})^\mathfrak{g} \to U(\mathfrak{g})^\mathfrak{g}\), given by an explicit formula, where \(S(\mathfrak{g})^\mathfrak{g}\) and \(U(\mathfrak{g})^\mathfrak{g}\) are the \(\mathfrak{g}\)-invariant subspaces for the adjoint action of \(\mathfrak{g}\) on the symmetric algebra and the universal enveloping algebra of \(\mathfrak{g}\), respectively. To show Duflo's theorem that the map \(\mathcal{D}\) is an algebra isomorphism, the difficulty is the part to show that \(\mathcal{D}\) is an algebra homomorphism, namely that \(\mathcal{D}\) is multiplicative. There have been many proofs of Duflo's theorem. A topological proof was given by the first author, Le and Thurston, for metrized Lie algebras, using the Kontsevich integral and giving an interpretation of ``\(1+1=2\) on an abacus'' in terms of knots in 3-dimensional space [\textit{D. Bar-Natan} et al., Geom. Topol. 7, 1--31 (2003; Zbl 1032.57008)]. In this paper, the authors give a proof of Duflo's theorem, for arbitrary finite-dimensional Lie algebras, using a ``4-dimensional abacus''.
The proof is given in such a process as follows. They use the setup and results due to the first and second authors [Math. Ann. 367, No. 3-4, 1517--1586 (2017; Zbl 1362.57005)]. They consider a certain circuit algebra called the space of w-foams such that each generator and relation is interpreted in terms of certain knotted objects in \(\mathbb{R}^4\) that are ribbon knotted tubes with foam vertices and strings in \(\mathbb{R}^4\). A 4-dimensional abacus bead is an element of the space of w-foams. They give an interpretation of ``\(1+1=2\)'' in terms of w-foams, and using a certain filtered linear map called the homomorphic expansion, they describe ``\(1+1=2\)'' diagrammatically in terms of arrow diagrams. Using the tensor interpretation map, they give an equality in \(\hat{S}(\mathfrak{g}^*)_\mathfrak{g} \otimes \hat{U}(\mathfrak{g})\). Here \(S(\mathfrak{g}^*)\) denotes the symmetric algebra of the linear dual of \(\mathfrak{g}\), and \(\hat{}\) denotes the degree completion where elements of \(\mathfrak{g}^*\) and \(\mathfrak{g}\) are defined to be degree 1 and degree 0, respectively, and the subscript \(\mathfrak{g}\) denotes co-invariants under the co-adjoint action of \(\mathfrak{g}\), and \(U(\mathfrak{g})\) and \(\hat{}\) denote the universal enveloping algebra of \(\mathfrak{g}\) and the degree completion, respectively. Then they show the Duflo's theorem. Further, they derive the explicit formula for the Duflo map from the homomorphic expansion.Periodic projections of alternating knotshttps://www.zbmath.org/1472.570022021-11-25T18:46:10.358925Z"Costa, Antonio F."https://www.zbmath.org/authors/?q=ai:costa.antonio-felix"Quach-Hongler, Cam Van"https://www.zbmath.org/authors/?q=ai:van-quach-hongler.camThis paper proves that for \(q\) greater than 2 every \(q\)-periodic prime alternating knot admits an alternating projection for which its \(q\)-periodicity is visible as a rotation of the knot diagram [Visibility Theorem 3.1], demonstrating as an application that the crossing number of such a knot is always a multiple of \(q\). It also shows, in an elementary way that does not depend on computer calculations, that knot 12a634 [Figure 24] is not \(q\)-periodic (an example which, this reviewer notes, has multiple 3-colorings with visible linking numbers between the branch curves of its corresponding 3-fold non-cyclic covering spaces, when projected in convenient ways).
For earlier work on periodicity, using covering linking invariants, see \textit{K. Murasugi}'s paper [Tsukuba J. Math. 4, 331--347 (1980; Zbl 0469.57002)].A diagrammatic approach for determining the braid index of alternating linkshttps://www.zbmath.org/1472.570032021-11-25T18:46:10.358925Z"Diao, Yuanan"https://www.zbmath.org/authors/?q=ai:diao.yuanan"Ernst, Claus"https://www.zbmath.org/authors/?q=ai:ernst.claus"Hetyei, Gabor"https://www.zbmath.org/authors/?q=ai:hetyei.gabor"Liu, Pengyu"https://www.zbmath.org/authors/?q=ai:liu.pengyuThe authors use two well-known inequalities regarding the braid index to derive explicit formulas for the braid index for alternating links. One inequality gives an upper bound on the braid index and the other one that is called the Morton-Frank-Williams inequality gives a lower bound on the braid index. The main idea of this paper is based on constructing a diagram of the given link for which the upper and lower bounds of the braid index are equal in these two inequalities. As an example, these formulas are applicable to all two bridge links, all alternating pretzel links and all alternating Montesinos links.Cohomology with local coefficients and knotted manifoldshttps://www.zbmath.org/1472.570042021-11-25T18:46:10.358925Z"Ellis, Graham"https://www.zbmath.org/authors/?q=ai:ellis.graham-j"Killeen, Kelvin"https://www.zbmath.org/authors/?q=ai:killeen.kelvinThis paper shows how classical ideas used on knot theory can be encoded in a way that makes it possible to use a computer to calculate ambient isotopy invariants of continuous embeddings \(N \ \hookrightarrow \ M\). The authors describe an algorithm for computing the homology and cohomology of a finite connected CW-complex \(X\) with coefficients in a \({\mathbb Z}\pi_{1}(X)\) module \(A\) when \(A\) is finitely generated over \(\mathbb Z\). As examples of the effectiveness of the algorithm, which is implemented in the Groups, Algorithms, and Programming system GAP (\url{http://www.gap-system.org}) and in HAP -- the homological homological algebra programming system (\url{http://hamilton.nuigalway.ie/}) built on GAP -- the authors give two illustrations of the technique. The first shows that degree 2 homology distinguishes the homotopy types of the complements of the spun Hopf link and Satoh's tube map of the welded Hopf links. The second example shows that the system distinguishes between the homeomorphism types of the complements of the granny knot and the reef knot. The details of the implementations are given in the paper. In order to make the computations manageable, a CW complex \(X\) is represented by a regular CW complex \(Y\) -- that is one whose attaching maps restrict to homeomorphisms on cell boundaries -- together with a simple homotopy equivalence \(Y \simeq X\). The paper contains many useful diagrams to illustrate the constructions used. Timings for the execution of the various codes are given based on execution on a standard GNU/Linux box. Note that GAP/HAP has a substantial number of built-in routines so that the algorithms illustrated in the paper are quite short.An \(\mathfrak{sl}_n\) stable homotopy type for matched diagramshttps://www.zbmath.org/1472.570052021-11-25T18:46:10.358925Z"Jones, Dan"https://www.zbmath.org/authors/?q=ai:jones.dan"Lobb, Andrew"https://www.zbmath.org/authors/?q=ai:lobb.andrew"Schütz, Dirk"https://www.zbmath.org/authors/?q=ai:schutz.dirkKhovanov homology \(\mathrm{Kh}^{*,*}(K)\) is a bi-graded vector space associated to the knot \(K\). In~the paper [J. Am. Math. Soc. 27, No. 4, 983--1042 (2014; Zbl 1345.57014)], \textit{R. Lipshitz} and \textit{S. Sarkar} defined a {stable Khovanov homotopy type} associated to a knot \(K\); that is to say, a family \(\{\mathcal{X}^{j}(D_K) \}_{j\in \mathbb{Z}}\) of topological spaces -- defined combinatorially starting from a diagram \(D_K\) of \(K\) -- such that: (i) for each \(j\) the space \(\mathcal{X}^{j}(K)\) does not depend up to homotopy on the choice of \(D_K\), and (ii) the reduced singular cohomology of \(\mathcal{X}^{j}(K)\) is isomorphic to \(\mathrm{Kh}^{*,j}(K)\). One important feature of this construction is that it allows one to import part of the algebraic structure of singular cohomology in Khovanov homology. More precisely, all stable cohomology operations, such as Steenrod squares, give rise to knot invariants. Indeed, in [loc. cit.] it has been shown that these Steenrod squares are computable and can be used to prove that the homotopy type of \(\{\mathcal{X}^{j}(D_K) \}_{j\in \mathbb{Z}}\) is a stronger invariant than \(\mathrm{Kh}^{*,*}(K)\). Lipshitz and Sarkar's construction is based on flow categories and based on the template given by \textit{R. L. Cohen} et al. [Prog. Math. 133, 297--325 (1995; Zbl 0843.58019)].
Khovanov homology fits into a family of theories called Khovanov-Rozansky \(\mathfrak{sl}_n\)-homologies (Khovanov homology being the case \(n=2\)). The aim of the paper under review is to extend the definition of stable Khovanov homotopy type to Khovanov-Rozansky \(\mathfrak{sl}_n\)-homologies. This is achieved only for a certain class of knots, called bipartite, which admit special (matched) diagrams. Bipartite knots include rational knots and an infinite family of Montesinos links. However, not all links are bipartite. For bipartite diagrams the description of the Khovanov-Rozansky \(\mathfrak{sl}_n\) chain complex can be simplified, and made similar to the description of the Khovanov complex. Making use of this similarity, the authors define a framed flow category, and thus a stable homotopy type, associated to each pair consisting of a matched diagram and of an integer \(n\geq 2\). In the case \(n = 2\), their construction agrees with Lipshitz-Sarkar's construction. For all \(n\geq3\), the cohomology of the stable homotopy type agrees with the Khovanov-Rozansky \(\mathfrak{sl}_n\) homology of the underlying knot. Moreover, the authors prove that their stable homotopy type is preserved under some Reidemeister-like moves which send matched diagrams to matched diagrams. The last section of the paper under review is dedicated to examples and calculations.Primary decompositions of knot concordancehttps://www.zbmath.org/1472.570062021-11-25T18:46:10.358925Z"Livingston, Charles"https://www.zbmath.org/authors/?q=ai:livingston.charlesA central problem in three-dimensional knot concordance theory in the smooth category consists of understanding \(\mathcal{T}\), the concordance group of topologically slice knots.
In this paper, the author shows that primary decompositions of \(\mathcal{T}\) of a strong type cannot exist. The main goal of this paper is to provide a counterexample to a splitting property related to the surjectivity of \(\Phi\), considered by Cha under the name strong existence [\textit{J. C. Cha}, Forum Math. Sigma 9, Paper No e57, 37 p. (2021; Zbl 07387780)]. The main theorems are listed as follows.
\textbf{Theorem 1.1} There exists a set of three polynomials , \(\mathcal{P}_0=\{f_1,f_2,f_3\}\subset \mathcal{P}\), such that the natural homomorphism \[ \mathcal{T}_{\Delta}^{f_1}\oplus \mathcal{T}_{\Delta}^{f_2}\oplus\mathcal{T}_{\Delta}^{f_3}\rightarrow \mathcal{T}_{\Delta}^{\mathcal{P}_0} \] is not surjective.
\textbf{Theorem 1.2} There exist Alexander polynomials \(f_1(t)\) and \(f_2(t)\) having no common factors and a topologically slice knot \(K\) with \(\Delta_K(t)=f_1(t)^2f_2(t)^2\) such that \(K\) is not concordant to any connected sum of knots \(K_1\sharp K_2\) where \(\Delta_{K_i}(t)=f_i(t)^{n_i}\) and \(n_1,n_2\in \mathbb{Z}\).
It follows that \(K\) is a topologically slice knot that is not smoothly concordant to a knot with Alexander polynomial one. The first examples of such knots were described in [\textit{M. Hedden} et al., Adv. Math. 231, No. 2, 913--939 (2012; Zbl 1254.57008)].
In the appendix of this paper, the author also describes the computation of the \(d\)-invariants for the 2-fold branched cover of \(S^3\) branched over \(K_{15}\).Some Baumslag-Solitar groups are two-bridge virtual knot groupshttps://www.zbmath.org/1472.570072021-11-25T18:46:10.358925Z"Mira-Albanés, Jhon Jader"https://www.zbmath.org/authors/?q=ai:mira-albanes.jhon-jader"Rodríguez-Nieto, José Gregorio"https://www.zbmath.org/authors/?q=ai:rodriguez-nieto.jose-gregorio"Salazar-Díaz, Olga Patricia"https://www.zbmath.org/authors/?q=ai:salazar-diaz.olga-patriciaGroup presentations for links in thickened surfaceshttps://www.zbmath.org/1472.570082021-11-25T18:46:10.358925Z"Silver, Daniel S."https://www.zbmath.org/authors/?q=ai:silver.daniel-s"Williams, Susan G."https://www.zbmath.org/authors/?q=ai:williams.susan-gGiven a standard link diagram, there are two well-known methods of constructing a group presentation for the fundamental group of the link complement. The Wirtinger presentation has generators corresponding to overcrossing arcs of the diagram, while the Dehn presentation has generators corresponding to regions of the diagram. Both presentations have relations determined by the crossings of the diagram. The first part of this paper gives an explicit isomorphism between the two groups in terms of the generators in the presentations. The authors frame the construction as a form of `integration' along paths in the link diagram.
Next the paper moves to the more general setting of a link in a thickened (connected, closed, compact, orientable) surface. In this setting the Wirtinger and Dehn group presentations can still be defined, but they might not give the fundamental group of the link complement and they might not be equal. The authors again use the `integration' to explicitly set out the relationship between the two groups.
The second half of the article considers the Dehn colouring group of a link in a thickened surface that can be checkerboard shaded. This is defined using Fox's free differential calculus along with the relationship just described. The two possible checkerboard shadings give rise to two (weighted) Tait graphs, and the adjacency matrices of these graphs give the Laplacian groups. This pair of groups has previously been shown to be a link invariant. The authors explore the relationship between these groups and the Dehn colouring group.
Using homological information about the surface, these invariants are then extended to modules. The two Laplacian modules each provide a Laplacian polynomial. The final result of this paper is that, in the case where the surface is a torus, the two polynomials are equal.tg-hyperbolicity of virtual linkshttps://www.zbmath.org/1472.570092021-11-25T18:46:10.358925Z"Adams, Colin"https://www.zbmath.org/authors/?q=ai:adams.colin-c"Eisenberg, Or"https://www.zbmath.org/authors/?q=ai:eisenberg.or"Greenberg, Jonah"https://www.zbmath.org/authors/?q=ai:greenberg.jonah"Kapoor, Kabir"https://www.zbmath.org/authors/?q=ai:kapoor.kabir"Liang, Zhen"https://www.zbmath.org/authors/?q=ai:liang.zhen"O'Connor, Kate"https://www.zbmath.org/authors/?q=ai:oconnor.kate"Pacheco-Tallaj, Natalia"https://www.zbmath.org/authors/?q=ai:pacheco-tallaj.natalia"Wang, Yi"https://www.zbmath.org/authors/?q=ai:wang.yi.10|wang.yi.5|wang.yi.9|wang.yi.3|wang.yi.8|wang.yi.7|wang.yi.4|wang.yi.1|wang.yi.6Let \(M\) be a compact oriented \(3\)-manifold. Let \(M'\) be the \(3\)-manifold obtained from \(M\) by deleting any genus one components of \(\partial M\) and capping off any \(2\)-sphere components of \(\partial M\) with \(3\)-balls. Then \(M\) is said to be \emph{tg-hyperbolic} if \(M'\) admits a hyperbolic metric of finite volume that is totally geodesic on all components of \(\partial M'\). If \(\Sigma\) is a closed oriented surface and \(\mathscr{L} \subset \Sigma \times [0,1]\) is a link in \(\Sigma \times [0,1]\), then \(\mathscr{L}\) is said to be \emph{tg-hyperbolic} if \(\Sigma \times I \smallsetminus \mathscr{L}\) is tg-hyperbolic. Every virtual link diagram \(L\) can be represented as a link \(\mathscr{L} \subset \Sigma \times [0,1]\). If \(L\) has a representative \(\mathscr{L} \subset \Sigma \times [0,1]\) that is tg-hyberbolic, then the virtual link \(L\) is also called tg-hyperbolic.
By a theorem of Kuperberg, every virtual link \(L\) can be represented as a link \(\mathscr{L} \subset \Sigma \times [0,1]\) that is minimal in the following sense: every vertical annulus in \(\Sigma \times [0,1]\) that is disjoint from \(\mathscr{L}\) intersects \(\Sigma \times \{1\}\) in an inessential circle. The smallest genus of all surfaces on which a virtual link can be represented is called its virtual genus. Then it follows from a theorem of Thurston that if \(L\) is a tg-hyperbolic virtual link and \(\mathscr{L} \subset \Sigma \times [0,1]\) is a tg-hyperbolic representative of \(L\), then the genus of \(\Sigma\) is the virtual genus of \(L\).
The authors of the paper under review provide many useful criteria for showing when a virtual link is tg-hyperbolic. They show that if a virtual link \(L\) of virtual genus at least 2 has a minimal genus representative that does not admit any essential tori, then \(L\) is tg-hyperbolic. As a consequence, they show that the generalized Kishino knot \(K_n\) is hyperbolic for every \(n \ge 2\). It is also shown that if \(L\) has a non-trivial alternating virtual link diagram having a prime minimal genus representative and if \(L\) is not a classical \(2\)-braid, then \(L\) is tg-hyperbolic.
Lastly, the authors study the volumes of tg-hyperbolic links. Among many interesting results in this direction, they show that the minimal volume tg-hyperbolic 2-component virtual link is exactly \(2 \cdot v_{\text{oct}}\). For virtual knots, they conjecture that the virtual trefoil realizes the minimal volume (\(\approx 5.33349\)). The paper concludes with a table of hyperbolic volumes for the 116 non-trivial virtual knots up to four classical crossings. Of these, only the classical trefoil is not tg-hyperbolic.Welded extensions and ribbon restrictions of diagrammatical moveshttps://www.zbmath.org/1472.570102021-11-25T18:46:10.358925Z"Colombari, Boris"https://www.zbmath.org/authors/?q=ai:colombari.borisA string link is an embedding of several copies of an interval \(I\) in \(B^2 \times I\) with ends fixed in \(B^2 \times \partial I\). A virtual string link diagram is a generic projection of a string link on a plane such that it has classical crossings and virtual crossings. A welded string link is an equivalence class of virtual string link diagrams under local moves called welded Reidemeister moves. A classical local move is a local move which only involves classical crossings. For \(n\)-component virtual string link diagrams, when a classical move \(M_c\) can be realized using a local move \(M_w\) and welded Reidemeister moves, we say that \(M_w\) is a welded extension of \(M_c\) if the induced map \(\mathcal{SL}_n^{M_c} \to w\mathcal{SL}_n^{M_w}\) is injective, where \(\mathcal{SL}_n^{M_c} \) is the set of equivalence classes of string links under \(M_c\) and \(w\mathcal{SL}_n^{M_w}\) is the set of equivalence classes of welded string links under \(M_w\), respectively. A string link is related to a ribbon string 2-link, which is a string 2-link, an embedding of several copies of \(S^1 \times I\) in \(B^3 \times I\) with boundaries fixed in \(B^3 \times \partial I\), with only ribbon singularities, and there are two maps called the Tube map and the Spun map that give the relation. \newline The main results are as follows. Considering local moves and string ribbon 2-links given by the Tube map and the Spun map, the author gives the notion that a local move \(M_w\) is a ribbon residue of a classical local move \(M_c\), as a method to study welded extensions. The author considers specifically three local moves: the self-crossing change, the Delta move, and the band-pass move. For the self-crossing change, a welded extension called the self-virtualization move is a ribbon residue. For the Delta move, there are welded extensions called the fused move \(F\) and the virtual conjugation move VC; the author shows that \(F\) is a ribbon residue while VC is not. For the band-pass move, the virtualization move V, which replaces a classical crossing with a virtual crossing, is a ribbon residue, that is not a welded extension. For this case, the author considers a move called the band virtualization move BV derived from the band-pass move, and shows that BV is in some sense a ``restricted ribbon residue'' of the band pass move.Knot polynomials of open and closed curveshttps://www.zbmath.org/1472.570112021-11-25T18:46:10.358925Z"Panagiotou, Eleni"https://www.zbmath.org/authors/?q=ai:panagiotou.eleni"Kauffman, Louis H."https://www.zbmath.org/authors/?q=ai:kauffman.louis-hirschSummary: In this manuscript, we introduce a method to measure entanglement of curves in 3-space that extends the notion of knot and link polynomials to open curves. We define the bracket polynomial of curves in 3-space and show that it has real coefficients and is a continuous function of the curve coordinates. This is used to define the Jones polynomial in a way that it is applicable to both open and closed curves in 3-space. For open curves, the Jones polynomial has real coefficients and it is a continuous function of the curve coordinates and as the endpoints of the curve tend to coincide, the Jones polynomial of the open curve tends to that of the resulting knot. For closed curves, it is a topological invariant, as the classical Jones polynomial. We show how these measures attain a simpler expression for polygonal curves and provide a finite form for their computation in the case of polygonal curves of 3 and 4 edges.The third term in lens surgery polynomialshttps://www.zbmath.org/1472.570122021-11-25T18:46:10.358925Z"Tange, Motoo"https://www.zbmath.org/authors/?q=ai:tange.motooSummary: It is well-known that the second highest coefficient of the Alexander polynomial of any lens space knot in \(S^3\) is \(-1\). We show that if the third highest coefficient of the Alexander polynomial \(\Delta_K(t)\) of a lens space knot \(K\) in \(S^3\) is non-zero, then \(\Delta_K(t)\) coincides with the Alexander polynomial of the \((2,2g+1)\)-torus knot, where \(g\) is the Seifert genus of \(K\).A diagrammatic approach to the AJ conjecturehttps://www.zbmath.org/1472.570132021-11-25T18:46:10.358925Z"Detcherry, Renaud"https://www.zbmath.org/authors/?q=ai:detcherry.renaud"Garoufalidis, Stavros"https://www.zbmath.org/authors/?q=ai:garoufalidis.stavrosThe AJ-conjecture relates the colored Jones polynomial with the character variety of a knot. The sequence of colored Jones polynomials of a knot \(K\) is \(q\)-holonomic, namely it satisfies a linear recursive relation. This relation is encoded by a non-commutative polynomial on 3-variables \(\hat A_K\). On the other hand the \(A\)-polynomial \(A_K\) is the two variable polynomial defined by the variety of characters of \(K\) restricted to the peripheral group, after removing the abelian component. The result of replacing \(q=1\) at \(\hat A_K\) is a commutative 2-variable polynomial; the AJ conjecture claims that this polynomial and \(A_k\) have the same irreducible factors (after some change of variable), namely they define the same planar curve.
The current paper approaches the AJ-conjecture by means of diagrams. The authors define a version of the \(\hat A\)-polynomial that could depend on a diagram \(D\), \(\hat A^c_D\), also a non-commutative polynomial in 3-variables. It is conjectured that \(\hat A^c_D\) does not depend on the diagram but just on the knot \(K\). As it is not yet known that \(\hat A^c_D\) is independent of the diagram, \(\hat A^c_K\) is defined as the left gcd of all \(\hat A^c_D\) for all diagrams \(D\) representing \(K\). The main result is that (a) \(\hat A_K\) divides \(\hat A^c_K\), and (b) irreducible factors of the \(A\)-polynomial \(A_K\) are also irreducible factors of the result of replacing \(q=1\) at \(\hat A^c_K\) (after some change of variable).
The proof builds on the octahedral decomposition of the knot complement minus two points from a diagram \(D\), and uses a theorem proved by \textit{H. Kim} et al. [Geom. Dedicata 197, 123--172 (2018; Zbl 1411.57002)] namely that the gluing equations of these octahedra yield \(A_K\), the \(A\)-polynomial of the knot \(K\).Some unitary representations of Thompson's groups \(F\) and \(T\)https://www.zbmath.org/1472.570142021-11-25T18:46:10.358925Z"Jones, Vaughan F. R."https://www.zbmath.org/authors/?q=ai:jones.vaughan-f-rSummary: In a ``naive'' attempt to create algebraic quantum field theories on the circle, we obtain a family of unitary representations of Thompson's groups \(T\) and \(F\) for any subfactor. The Thompson group elements are the ``local scale transformations'' of the theory. In a simple case the coefficients of the representations are polynomial invariants of links. We show that all links arise and introduce new ``oriented'' subgroups of \(\overrightarrow{F} < F\) and \(\overrightarrow{T} < T\) which allow us to produce all oriented knots and links.On sign assignments in link Floer homologyhttps://www.zbmath.org/1472.570152021-11-25T18:46:10.358925Z"Eftekhary, Eaman"https://www.zbmath.org/authors/?q=ai:eftekhary.eamanSummary: In this short note, we compare the combinatorial sign assignment of Manolescu, Ozsváth, Szabó, and Thurston [\textit{C. Manolescu} et al., Geom. Topol. 11, 2339--2412 (2007; Zbl 1155.57030)] for grid homology of knots and links in \(S^3\) with the sign assignment coming from a coherent system of orientations on Whitney disks [\textit{A. S. Alishahi} and \textit{E. Eftekhary}, J. Symplectic Geom. 13, No. 3, 609--743 (2015; Zbl 1335.53112)]. Although these constructions produce different signs, a small modification of the convention in either of the two methods results in identical sign assignments, and thus identical chain complexes.A Thurston boundary for infinite-dimensional Teichmüller spaceshttps://www.zbmath.org/1472.570162021-11-25T18:46:10.358925Z"Bonahon, Francis"https://www.zbmath.org/authors/?q=ai:bonahon.francis"Šarić, Dragomir"https://www.zbmath.org/authors/?q=ai:saric.dragomirThe Teichmüller space of a Riemann surface \(X_0\) is the space of quasiconformal deformations of the complex structure of \(X_0\). Thurston compactified the Teichmüller space of a compact Riemann surface \(X_0\), of genus at least 2, by adding a boundary at infinity consisting of projective measured foliations or, equivalently, projective measured geodesic laminations. In the present paper, the authors introduce a similar construction of a boundary for the Teichmüller space of a noncompact surface \(X_0\), using the technical tool of geodesic currents. ``In addition to the fact that Teichmüller spaces of noncompact Riemann surfaces are fundamental objects in complex analysis, our motivation here is to put in evidence the hidden features that underlie Thurston's construction, by tying it more closely to the quasiconformal geometry of \(X_0\) and less to the purely topological considerations that suffice for compact surfaces.''Non virtually solvable subgroups of mapping class groups have non virtually solvable representationshttps://www.zbmath.org/1472.570172021-11-25T18:46:10.358925Z"Hadari, Asaf"https://www.zbmath.org/authors/?q=ai:hadari.asafLet \(\Sigma\) denote a compact orientable surface of finite type with at least one boundary component. Let Mod\((\Sigma)\) and \(\Gamma\) denote the mapping class group of \(\Sigma\) and a non virtually solvable subgroup of Mod\((\Sigma)\), respectively. The author points out that A. Lubotzky asked the following question: Given a non virtually solvable subgroup \(\Gamma < \mathrm{Mod}(\Sigma)\), is there a finite dimensional representation \(\rho : \mathrm{Mod}(\Sigma) \to GL(V)\) such that \(\rho(\Gamma)\) is not virtually solvable?
The author establishes an affirmative answer to this question, provided the surface has non-empty boundary. More explicitly, the author shows that there is a finite cover \(p: \Sigma' \to \Sigma\) such that the image \(\rho_p (\Gamma)\) under the homological representation \(\rho_p\) is not virtually solvable. Moreover, if \(\Gamma\) contains a pseudo-Anosov element then this cover can be taken to be a regular cover with a solvable deck group.
Using \textit{A. Lubotzky} and \textit{C. Meiri}'s result which is one of the main results in their paper [J. Am. Math. Soc. 25, No. 4, 1119--1148 (2012; Zbl 1283.20075)], together with the result above, the author obtains the following two corollaries related to random elements in subgroups of Mod\((\Sigma)\):
Let \(\Sigma\) be as above and let \(\Gamma < \mathrm{Mod}(\Sigma)\) be a non virtually solvable finitely generated group.
1. Then \(\bigcup_{m=2}^{\infty} \Gamma^{m}\) is exponentially small in \(\Gamma\).
2. Let \(X \subset \Gamma\) be the set of all elements with topological entropy 0. Then \(X\) is exponentially small in \(\Gamma\).
The author also notes that the last corollary is already known (see for example [\textit{M. T. Clay} et al., Groups Geom. Dyn. 6, No. 2, 249--278 (2012; Zbl 1245.57004); \textit{J. Maher}, J. Lond. Math. Soc., II. Ser. 86, No. 2, 366--386 (2012; Zbl 1350.37010); \textit{T. Sakasai}, IRMA Lect. Math. Theor. Phys. 17, 531--594 (2012; Zbl 1272.57001)]) and has been proved by several authors using different methods.Small asymptotic translation lengths of pseudo-Anosov maps on the curve complexhttps://www.zbmath.org/1472.570182021-11-25T18:46:10.358925Z"Kin, Eiko"https://www.zbmath.org/authors/?q=ai:kin.eiko"Shin, Hyunshik"https://www.zbmath.org/authors/?q=ai:shin.hyunshikLet \(S : = S_{g,n}\) be an orientable surface of genus \(g\) with \(n\) punctures. The curve complex \(\mathcal{C}(S)\) consists of the homotopy classes of essential simple close curves on \(S\) as vertices and the 1-simplices connecting disjoint pairs of such curve classes. The vertex set of \(\mathcal{C}(S)\) becomes a metric space (with metric \(d_{\mathcal{C}}\)) by assigning distance 1 to each 1-simplex. Let \(\mathrm{Mod}(S)\) be the mapping class group of \(S\). Then \(\mathrm{Mod}(S)\) acts naturally on \(\mathcal{C}(S)\), preserving \(d_{\mathcal{C}}\). Let \(\alpha \subset S\) be a simple essential curve. The asymptotic translation length of \(f \in \mathrm{Mod}(S)\) is \[ l_{\mathcal{C}}(f) = \lim_{j \to \infty} \inf \frac{d_{\mathcal{C}}(\alpha, f^j(\alpha))}{j}. \] \(l_{\mathcal{C}}(f)\) is independent of the choice of \(\alpha\), hence, is an invariant of \(f\). Moreover, \(l_{\mathcal{C}}(f) > 0\) if and only if \(f\) is pseudo-Anosov. Suppose \(H \le \mathrm{Mod}(S)\). Then \(L_{\mathcal{C}}(H)\), the minimum of \(l_{\mathcal{C}(f)}\) over pseudo-Anosov \(f \in H\), is an invariant of \(H\).
\(L_{\mathcal{C}}(\mathrm{Mod}(S))\) is always finite and, more importantly, is bounded below and away from zero. This means the same are true for \(L_{\mathcal{C}}(H)\) with \(H < \mathrm{Mod}(S)\). The specific bounds depend only on \(g,n\) and are related to the Euler characteristic of \(S\). The present paper gives detailed and explicit computations on various special cases of \(g, n\) and improves on the bounds.
Let \(\psi\) be pseudo-Anosov. Suppose there is an essential simple closed curve \([b] \in \mathrm{H}_1(S,\mathbb{Z})\) such that \(\psi[b] = [b]\). Let \(p : \tilde{S} \to S\) be the \(\mathbb{Z}\)-cover corresponding to \([b] \in \pi_1(S)\) and \(h : \tilde{S} \to \tilde{S}\) be a generator of the corresponding deck transformation. Then there is a lift \(\tilde{\psi} : \tilde{S} \to \tilde{S}\). Let \(R_n = \tilde{S}/\langle h^n \tilde{\psi} \rangle\). Then \(h^{-1}\) descends to \(\psi_n \in \mathrm{Mod}(R_n)\). The main theorem (A) states that there exists a constant \(C\) such that for sufficiently large \(n\), \[ l_{\mathcal{C}}(\psi_n) \le \frac{C}{\chi(R_n)}. \]
The subject is motivated by mapping tori and 3-dimensional hyperbolic manifolds in general. The rest of the paper gives consequences and applications of the main theorem in this context. The authors prove and improve on the bounds on \(L_{\mathcal{C}}(\mathrm{Mod}(S_{1,g})\) and \(L_{\mathcal{C}}(S_{n,0})\) and \(L_{\mathcal{C}}(H)\) for subgroups \(H < \mathrm{Mod}(S)\), where \(H\) is either the hyperelliptic mapping class group or the handlebody subgroup over \(S_{g,0}\).Weil-Petersson translation length and manifolds with many fibered fillingshttps://www.zbmath.org/1472.570192021-11-25T18:46:10.358925Z"Leininger, Christopher"https://www.zbmath.org/authors/?q=ai:leininger.christopher-j"Minsky, Yair N."https://www.zbmath.org/authors/?q=ai:minsky.yair-n"Souto, Juan"https://www.zbmath.org/authors/?q=ai:souto.juan"Taylor, Samuel J."https://www.zbmath.org/authors/?q=ai:taylor.samuel-josephThe article under review studies the structure of pseudo-Anosov mapping classes with respect to their Weil-Petersson translation lengths. This translation length is obtained via the action of mapping classes on Teichmüller space equipped with the Weil-Petersson metric. The paper contains two main results, highlighting both the similarity and the difference to the structure of pseudo-Anosov mapping classes with respect to the Teichmüller translation length, which in turn is obtained via the action of mapping classes on Teichmüller space equipped with the Teichmüller metric.
A basic comparison of the Weil-Petersson translation length and the Teichmüller translation length is due to a result of \textit{M. Linch} [Proc. Am. Math. Soc. 43, 349--352 (1974; Zbl 0284.32015)]: the Weil-Petersson translation length of a pseudo-Anosov map~\(f: \Sigma \to \Sigma\) is bounded from above by the normalised Teichmüller translation length, where the normalisation is by the square root of the hyperbolic area of~\(\Sigma\).
For the Teichmüller translation length, the following finiteness properties are known to hold:
\begin{enumerate}
\item[(1)] on every surface~\(\Sigma\) and for every fixed constant~\(L>0\), there are finitely many conjugacy classes of pseudo-Anosov mapping classes with Teichmüller translation length at most~\(L\),
\item[(2)] pseudo-Anosov maps of bounded normalised Teichmüller translation length give rise to finitely many mapping tori (after removing a finite number of simple closed curves) by a result of \textit{I. Agol} and \textit{D. Margalit} [Adv. Math. 228, 1466--1502 (2011; Zbl 1234.37022)].
\end{enumerate}
The main results of the article concern the the Weil-Petersson translation length and provide a negative analogue to (1) and a positive analogue to (2).
More precisely, the first main result (Corollary 1.2) states that for large enough~\(L\), the set of all pseudo-Anosov mapping classes with normalised Weil-Petersson translation length at most~\(L\) contains infinitely many conjugacy classes of pseudo-Anosov maps on every closed surface of genus at least two. The proof goes via an extension (Theorem 1.1) of Linch's result to the composition of a pseudo-Anosov mapping class with the power of a Dehn twist along a suitable curve.
The second main result (Theorem 1.4) states that the mapping tori of pseudo-Anosov maps of bounded normalised Weil-Petterson translation length give rise to finitely many 3-manifolds after removing a finite number of simple closed curves. The proof uses a careful study of fibered fillings of hyperbolic complements of links in 3-manifolds (Theorem 1.5). Contrary to the situation with respect to the Teichmüller translation length, the construction removes also certain curves that are so-called level, which means they are contained in a single level set of the mapping torus of the original pseudo-Anosov map. Hence, the resulting 3-manifolds do not inherit a mapping torus structure.Efficient triangulations and boundary slopeshttps://www.zbmath.org/1472.570202021-11-25T18:46:10.358925Z"Bryant, Birch"https://www.zbmath.org/authors/?q=ai:bryant.birch"Jaco, William"https://www.zbmath.org/authors/?q=ai:jaco.william-h"Rubinstein, J. Hyam"https://www.zbmath.org/authors/?q=ai:rubinstein.j-hyamLet \(M\) be a compact, orientable, irreducible, boundary-irreducible, and an-annular 3-manifold. This paper proves by normal surface theory that \(M\) admits only finitely many boundary slopes for incompressible and boundary-incompressible surfaces of bounded Euler characteristic.
A triangulation of \(M\) is said to be annular-efficient if it is 0-efficient, i.e., the only normal disks are vertex-linking, and the only normal incompressible annuli are thin edge-linking, i.e, they are normally isotopic into an arbitrarily small regular neighborhood of an edge in the triangulation. The authors show that if \(\mathcal{T}\) is an annular-efficient triangulation of \(M\), then there are only finitely many boundary-slopes for connected normal surfaces in \(\mathcal{T}\) of a bounded Euler characteristic. It is thus enough to prove that any \(M\) admits an annular-efficient triangulation.
It has been shown in [\textit{W. Jaco} and \textit{J. H. Rubinstein}, J. Differ. Geom. 65, No. 1, 61--168 (2003; Zbl 1068.57023)] that any triangulation \(\mathcal{T}\) of \(M\) can be modified by the crushing operation to an ideal triangulation \(\mathcal{T}'\) of the interior of \(M\). The authors prove in this paper that \(\mathcal{T}'\) can be further modified to an end-efficient ideal triangulation \(\mathcal{T}^{\ast}\). Then they prove that \(\mathcal{T}^{\ast}\) can be modified by inflation, developed in [\textit{W. Jaco} and \textit{J. H. Rubinstein}, Adv. Math. 267, 176--224 (2014; Zbl 1304.57034)], to an annular-efficient triangulation of \(M\).Many Haken Heegaard splittingshttps://www.zbmath.org/1472.570212021-11-25T18:46:10.358925Z"Sisto, Alessandro"https://www.zbmath.org/authors/?q=ai:sisto.alessandroA closed connected oriented 3-manifold \(M\) is Haken if it is irreducible and contains an incompressible surface. The author gives a criterion for a genus \(g\geq 2\) Heegaard splitting of distance at least two to yield a Haken manifold. The main idea is to construct a (steady) surface from a (steady) path in the curve complex graph of the Heegaard surface that satisfies certain conditions and then to prove that the obtained surface is incompressible.
Given integers \(g, n\) and \(k\) with \(g,n \geq 3\), the author applies the criterion to construct hyperbolic Haken manifolds which are integer homology spheres with Heegaard genus \(g\), Heegaard distance \(n\) and Casson invariant \(n\). It is worth mentioning that these properties were not known to be achieved at the same time.On the asymptotic expansion of the quantum SU(2) invariant at \(\zeta=e^{4\pi i/r}\) for Lens space \(L(p,q)\)https://www.zbmath.org/1472.570222021-11-25T18:46:10.358925Z"Takata, Toshie"https://www.zbmath.org/authors/?q=ai:takata.toshie"Tanaka, Rika"https://www.zbmath.org/authors/?q=ai:tanaka.rikaThe authors study the asymptotic expansion of the quantum \(SU(2)\) invariant for the lens space \(L(p,q)\) at \(\zeta=e^{\frac{4\pi i}{r}}\) with odd \(r\). The quantum \(SU(2)\) invariant at \(\zeta=e^{\frac{4\pi i}{r}}\) with odd \(r\) was defined by \textit{C. Blanchet} et al. [Topology 34, No. 4, 883--927 (1995; Zbl 0887.57009)] following a skein-theoretic approach pioneered by Lickorish, extending the Reshetikhin-Turaev invariants associated to \(\zeta = e^{\frac{2\pi i}{r}}\). Asymptotic expansions for quantum \(SU(2)\) invariants at \(\zeta=e^{\frac{4\pi i}{r}}\) with odd \(r\) are closely related to \(SL(2;\mathbb{C})\) Chern-Simons theory, as it was recently observed by \textit{Q. Chen} and \textit{T. Yang} [Quantum Topol. 9, No. 3, 419--460 (2018; Zbl 1405.57020)] that the asymptotic expansion determines the hyperbolic volume of the 3-manifold. It was shown by \textit{T. Ohtsuki} and \textit{T. Takata} [Commun. Math. Phys. 370, No. 1, 151--204 (2019; Zbl 1441.57011)] that for some Seifert 3-manifolds those asymptotic expansions can be represented by a sum of contributions from \(SL(2;\mathbb{C})\) flat connections whose coefficients are square roots of the Reidemeister torsions.
The authors of this paper extend the analysis for lens spaces \(L(p,q)\), and confirm that the same is true for the lens spaces. The proof involves calculation of the quantum invariants for \(L(p,q)\) and explicit comparison of their asymptotic expansions with the square roots of Reidemeister torsions.Volumes of two-bridge cone manifolds in spaces of constant curvaturehttps://www.zbmath.org/1472.570232021-11-25T18:46:10.358925Z"Mednykh, A. D."https://www.zbmath.org/authors/?q=ai:mednykh.alexander-dThe author considers hyperbolic, Euclidean and spherical cone-manifolds $\Sigma(\alpha)$ whose underlying topological space is the 3-sphere and whose singular set, with a cone-angle $\alpha$ around it ($0 < \alpha \le 2\pi$), is a knot $\Sigma$ in $S^3$. If $\Sigma$ is a hyperbolic knot (i.e., has hyperbolic complement of finite volume) then, as a consequence of Thurston's hyperbolic Dehn surgery theorem, for all sufficiently small cone-angles of the form $\alpha = 2\pi/n$, the cone-manfold $\Sigma(\alpha)$ is also hyperbolic (in fact, a hyperbolic orbifold). After shortly discussing the situation for the trefoil and other torus knots, the first hyperbolic knot considered in the paper is the figure-8 knot $4_1$. Concerning cone-structures, the cone-manifold $4_1(\alpha)$ is hyperbolic for $0 < \alpha < 2\pi/3$, Euclidean for $\alpha = 2\pi/3$ (the 3-fold cyclic branched covering of the figure-8 knot is the Euclidean Hantzsche-Wendt manifold, cf. a paper by the reviewer in [Monatsh. Math. 110, No. 3--4, 321--327 (1990; Zbl 0717.57005)]), and spherical for $2\pi/3 < \alpha < 4\pi/3$. The first main results of the present paper are calculations of the volumes of these geometric cone-manifolds, and similar calculations are then done also for the other ten hyperbolic knots with at most seven crossings. All these knots are 2-bridge knots, so their 2-fold branched coverings are lens spaces and their cone-manifolds for $\alpha = \pi$ are spherical orbifolds (incidentally, with the only exception of the figure-8 knot, for all other hyperbolic knots the associated cone-manifolds (orbifolds) for $\alpha = 2\pi/n$ are hyperbolic, for $n \ge 3$, by the orbifold geometrization and Dunbar's list of the geometric, non-hyperbolic 3-orbifolds [\textit{W. D. Dunbar}, Rev. Mat. Univ. Complutense Madr. 1, No. 1--3, 67--99 (1988; Zbl 0655.57008)]).
``For 2-bridge knots with not more than 7 crossings, we present trigonometrical identities involving the lengths of singular geodesics and cone-angles of such cone-manifolds. Then these identities are used to produce exact integral formulae of the volume of the corresponding cone-manifold modeled in the hyperbolic, spherical and Euclidean geometries.'' Here the main tools for the volume calculations are the Schläfli formula and the $A$-polynomial for these manifolds. The paper serves also as a good survey on geometric cone-manifolds and orbifolds associated to knots, their volume-computations as well as on the relevant literature. (Question: Considering all hyperbolic 2-bridge knots, is the upper bound for hyperbolicity over all cone-angles equal to $\pi$?)Compact 4-manifolds admitting special handle decompositionshttps://www.zbmath.org/1472.570242021-11-25T18:46:10.358925Z"Casali, Maria Rita"https://www.zbmath.org/authors/?q=ai:casali.maria-rita"Cristofori, Paola"https://www.zbmath.org/authors/?q=ai:cristofori.paolaThe origin of the problem of this article is the following problem posed by Kirby -- does every simply-connected closed PL 4-manifold have a special handlebody decomposition, i.e., a handlebody decomposition without 1- and 3-handles? In this article, the authors detect a class of compact simply-connected PL 4-manifolds with empty or connected boundary, which admit such decompositions and, therefore, can be represented by (undotted) framed links. Moreover, this class includes any compact simply-connected PL 4-manifold with empty or connected boundary having colored triangulations that minimize the combinatorially defined PL invariants regular genus, gem-complexity or \(G\)-degree among all such manifolds with the same second Betti number.\( \mathbb{CP}^2\)-stable classification of 4-manifolds with finite fundamental grouphttps://www.zbmath.org/1472.570252021-11-25T18:46:10.358925Z"Kasprowski, Daniel"https://www.zbmath.org/authors/?q=ai:kasprowski.daniel"Teichner, Peter"https://www.zbmath.org/authors/?q=ai:teichner.peterLet \(M\) be a closed, connected \(4\)-manifold. The quadratic \(2\)-type \(Q_M:= (\pi_1 M, w_1M, \pi_2 M, k_M, \lambda_M)\) consists of the Postnikov \(2\)-type of \(M\), determined by \(\pi_1 M\),\(\pi_2 M\) (as a \(\pi_1 M\)-module) and the \(k\)-invariant in \(H^3(\pi_1 M; \pi_2 M)\), together with the orientation character \(w_1 M : \pi_1 M \to \{\pm 1\}\) and the equivariant intersection form \(\lambda_M : \pi_2 M \times \pi_2 M \to \mathbb{Z}[\pi_1 M]\).
Two \(4\)-manifolds are \(\mathbb{C}P^2\)-stably homeomorphic if they become homeomorphic after taking connected sum with finitely many copies of \(\mathbb{C}P^2\), where different numbers of copies for the two manifolds are allowed. Quadratic \(2\)-types of two manifolds are stably isomorphic if they become isomorphic after finitely many stabilizations as follows: \[ Q_M \to Q_{M\sharp \pm\mathbb{C}P^2}\cong (\pi_1 M, w_1 M, \pi_2 M \oplus\mathbb{Z}[\pi_1 M], (k_M, 0), \lambda_M \perp \langle\pm 1\rangle). \]
In this paper, the authors show that two closed, connected \(4\)-manifolds with finite fundamental groups are \(\mathbb{C}P^2\)-stably homeomorphic if and only if their quadratic \(2\)-types are stably isomorphic and their Kirby-Siebenmann invariants agree.Higher order corkshttps://www.zbmath.org/1472.570262021-11-25T18:46:10.358925Z"Melvin, Paul"https://www.zbmath.org/authors/?q=ai:melvin.paul-m"Schwartz, Hannah"https://www.zbmath.org/authors/?q=ai:schwartz.hannah-rThis paper shows that a finite number of exotic copies of a smooth, closed, simply-connected \(4\)-manifold \(X\) can be obtained by removing a cork from \(X\), and then regluing it by powers of a boundary diffeomorphism. Then this result is used to separate finite families of corks embedded in a fixed \(4\)-manifold.
From the introduction: ``A cork is a compact contractible \(4\)-manifold \(C\) equipped with a boundary diffeomorphism \(h: \partial C \rightarrow \partial C\). The cork \((C,h)\) is trivial if \(h\) extends to a diffeomorphism of \(C\); for example \((B^4,h)\) is trivial for any \(h\). The cork is finite of order \(n\) if \(h\) is periodic of order \(n\). Corks of order \(2\) will be called involutory.
If \(C\) is embedded in the interior of a \(4\)-manifold \(X\), then the associated cork twist
\[X_{C,h}:=(X-int(C))\cup_h C\]
is homeomorphic [\textit{M. H. Freedman}, J. Differ. Geom. 17, 357--453 (1982; Zbl 0528.57011)] but not generally diffeomorphic [\textit{S. Akbulut}, ibid. 33, No. 2, 335--356 (1991; Zbl 0839.57015)] to \(X\).''
The paper includes two main results: The Finite Cork Theorem (3.1) and the Separation Theorem (3.3).
\begin{enumerate}
\item Finite Cork Theorem: Let \(X_i\) (\(i\in\mathbb{Z}_n\)) be any finite list of compact simply-connected \(4\)-manifolds all homeomorphic to a given one \(X=X_0\) that is closed or bounded by a homology sphere. Then there is an AC cork \((C, h)\) of order \(n\) in \(X\), with simply-connected complement in the closed case, whose twists \(X_{C,h^i}\) are diffeomorphic to \(X_i\) for each \(i\in\mathbb{Z}_n\). In the bounded case, these diffeomorphisms can be chosen to extend any given boundary identifications \(f_i:\partial X\rightarrow\partial X_i\).
\item Separation Theorem: For any \(n\in\mathbb{N}\) and any family of corks \((C_1,\sigma_1),\dots, (C_n , \sigma_n)\) embedded in a closed simply-connected \(4\)-manifold \(X\), there is a corresponding family of simple involutory corks \((A_i,\tau_i)\), embedded disjointly and each with simply-connected complement in \(X\), whose twists \(X_{A_i,\tau_i}\) are diffeomorphic to \(X_{C_i ,\sigma_i}\) for each \(i\).
\end{enumerate}
For the proofs of these results, two technical lemmas are applied, namely the Relative Involutory Cork Theorem (1.16) and the Consolidation Theorem (2.1). The Finite Cork Theorem and the Relative Involutory Cork Theorem are the non-closed versions of similar theorems for closed manifolds; the former stated in [\textit{D. Auckly} et al., Algebr. Geom. Topol. 17, No. 3, 1771--1783 (2017; Zbl 1382.57010); \textit{M. Tange}, Int. J. Math. 28, No. 6, Article ID 1750034, 26 p. (2017; Zbl 1371.57025)] and the latter stated in [\textit{C. L. Curtis} et al., Invent. Math. 123, No. 2, 343--348 (1996; Zbl 0843.57020); \textit{R. Matveyev}, J. Differ. Geom. 44, No. 3, 571--582 (1996; Zbl 0885.57016)].
The tools and techniques of this paper include AC cobordisms, pinwheels and handle manipulations.Topological decompositions of the Pauli group and their influence on dynamical systemshttps://www.zbmath.org/1472.570272021-11-25T18:46:10.358925Z"Bagarello, Fabio"https://www.zbmath.org/authors/?q=ai:bagarello.fabio"Bavuma, Yanga"https://www.zbmath.org/authors/?q=ai:bavuma.yanga"Russo, Francesco G."https://www.zbmath.org/authors/?q=ai:russo.francesco-giuseppeIn this article, we will see that it is possible to realize a manifold which has the Pauli group \(P\) \(=\) \(\langle X, Y, Z \ | \ X^2 = Y^2 = Z^2 = 1, (YZ)^4 = (ZX)^4 = (XY)^4 = 1 \rangle\) as its fundamental group. The Pauli group \(P\) is a group of order \(16\) introduced by [\textit{W. Pauli jun.}, Z. Phys. 43, 601--623 (1927; JFM 53.0858.02)]. Specific expressions of the three generators are:
\[
X \\ = \\ \begin{bmatrix} 0&1 \\ 1&0 \end{bmatrix}, \\ Y \\ = \\ \begin{bmatrix} 0&-i\\ i&0 \end{bmatrix}\text{ and } Z \\ = \\ \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix}
\].
It is easy to check that \(X^2 = Y^2 = Z^2 = I\), where \(I\) is the identity matrix.
The authors' main result is the theorem below. Note that \(S^3\) indicates the three-sphere. Further, the quaternion group and a cyclic group of order four, denoted by \(\mathbb Q_8\) and \(\mathbb Z_4\) respectively, operate on \(S^3\). Both finite actions are isometries on \(S^3\), hence the quotient spaces \(U\) \(=\) \(S^3/Q_8\) and \(V\) \(=\) \(S^3/\mathbb Z_4\) are both elliptic three-manifolds. More specifically, \(U\) is a quaternion manifold which is a family of the prism manifolds and \(V\) is a lens space \(L(4, 1)\).
Theorem. There exist two compact path connected quotient spaces \(U\) \(=\) \(S^3/Q_8\) and \(V\) \(=\) \(S^3/\mathbb Z_4\) such that the following conditions are true:
\begin{enumerate}
\item \(U \cup V\) is a compact path connected space with \(U \cap V \neq \emptyset\), \(\pi_1(U \cap V) \cong \mathbb Z_2\) and
\(P \cong \pi_1(U \cup V) / N\) for some normal subgroup \(N\) of \(\pi_1(U \cup V)\).
\item \(U \# V\) is a Riemannian manifold of \(dim(U \# V) = 3\) and \(P \cong \pi_1(U \# V) / L\) for some normal subgroup \(L\) of \(\pi_1(U \# V)\), where \(\#\) denotes the usual connected sum of manifolds.
\end{enumerate}
In the first statement, the familiar assumptions of the Seifert and van Kampen Theorem are satisfied. For example, [\textit{C. Kosniowski}, A first course in algebraic topology. (1980; Zbl 0441.55001), Chapter 23] is a handy reference source regarding the theorem we are using here. In fact, its proof is a repeated use of the Seifert and van Kampen Theorem involving algebraic manipulations of elements in \(\pi_1(U)\) \(\cong\) \(Q_8\), \(\pi_1(V)\) \(\cong\) \(\mathbb Z_4\) and \(\pi_1(U \cap V)\) \(=\) \(\mathbb Z_2\) in order to obtain the appropriate relations.
Moreover, the article discusses connections with physics or quantum mechanics, with an emphasis on \textit{pseudo-bosons} and \textit{pseudo-fermions}.Lengths of closed geodesics on random surfaces of large genushttps://www.zbmath.org/1472.570282021-11-25T18:46:10.358925Z"Mirzakhani, Mariam"https://www.zbmath.org/authors/?q=ai:mirzakhani.maryam"Petri, Bram"https://www.zbmath.org/authors/?q=ai:petri.bramIn this paper, the authors consider the asymptotic behavior of the distribution of short closed geodesics on random hyperbolic spaces as the genera tend to infinity. The moduli space \(\mathcal{M}_g\) of Riemann surfaces of genus \(g\) admits a natural probability measure \(\mathbb{P}_g\) induced from the Weil-Petersson metric. For \(X\in \mathcal{M}_g\) and an interval \([a,b]\subset \mathbb{R}_+=\{x\in \mathbb{R}\mid x>0\}\), let \(N_{g,[a,b]}(X)\) denote the number of primitive closed geodesics on \(X\) with lengths in the given interval. For \(a_1<b_1\le a_2<b_2\le\) \(\cdots\) \(\le a_k<b_k\), let \((N_{[a_i,b_i]})_{i=1}^k\) be a vector of independent Poisson distributed random variables with means \(\lambda_{[a_i,b_i]}=\int_{a_i}^{b_i}(e^t+e^{-t}-2)/(2t)\, dt\) (\(1\le i\le k\)) on a probability space (with the probabilty measure \(\mathbb{P}\)) that is rich enough to carry such a variable.
The authors first show that for any \(m_i\in \mathbb{N}\) (\(i=1,\dots,k\)), \(\mathbb{P}_g(N_{g,[a_i,b_i]}=m_i, 1\le i\le k)\) tends to \(\mathbb{P}(N_{[a_i,b_i]}=m_i, 1\le i\le k)=\prod_{i=1}^k(\lambda_i^{m_i}e^{-\lambda_i}/m_i!)\) as \(g\to \infty\), where \(\lambda_i=\lambda_{[a_i,b_i]}\).
Let \(\mathrm{sys}(X)\) be the systole of \(X\in \mathcal{M}_g\) and \(\mathbb{E}_g(\mathrm{sys})\) the expectation. Using the relation \(\mathbb{P}_g(\mathrm{sys}\le x)=1-\mathbb{P}_g(N_{g,[0,x]}=0)\), the authors show that \(\mathbb{E}_g(\mathrm{sys})\) tends to \(\int_0^\infty e^{-\lambda_{[0,R]}}dR=1.61498\ldots\) as \(g\to \infty\). In the comparison with Brooks and Makover's result [\textit{R. Brooks} and \textit{E. Makover}, J. Differ. Geom. 68, No. 1, 121--157 (2004; Zbl 1095.30037)], the authors also notice by applying the above result that \(\mathbb{P}_g(\mathrm{sys}\ge b)\to e^{-\lambda_{[0,b]}}=0.339043\ldots\) as \(g\to \infty\) for \(b=2\cdot\cosh^{-1}(3/2)\), while the probability measure from Brooks and Makover's model asymptotically concentrates in the \(b\)-thick part of the moduli space.
In the appendix, the authors also give a sketch of an unpublished result by M. Mirzakhani that there is a universal constant \(A\), \(B>0\) so that for any sequence \(\{c_g\}_g\) of positive numbers with \(c_g<A\log g\), \(\mathbb{P}_g(\mathrm{sys}\ge c_g)<Bc_ge^{-c_g}\).Tetrahedral Coxeter groups, large group-actions on 3-manifolds and equivariant Heegaard splittingshttps://www.zbmath.org/1472.570292021-11-25T18:46:10.358925Z"Zimmermann, Bruno P."https://www.zbmath.org/authors/?q=ai:zimmermann.bruno-pThe paper considers finite group-actions on closed 3-manifolds \(M\) which preserve the two handlebodies of some Heegaard splitting of \(M\). Four types of actions are distinguished depending on the orientability of \(M\); whether the action is orientation-preserving; whether the action exchanges the handlebodies; the maximal order of a possible group-action; and whether the action of the subgroup that preserves the handlebodies is orientation-preserving.
The maximal possible orders of groups acting in these four ways are discussed as well as a hierarchy for finite group-actions on 3-manifolds.
Various examples of 3-manifolds and associated group-actions are given. In particular, tetrahedral Coxeter groups and large group-actions.Fake 13-projective spaces with cohomogeneity one actionshttps://www.zbmath.org/1472.570302021-11-25T18:46:10.358925Z"He, Chenxu"https://www.zbmath.org/authors/?q=ai:he.chenxu"Rajan, Priyanka"https://www.zbmath.org/authors/?q=ai:rajan.priyankaFake real projective spaces are manifolds that are homotopy equivalent but not diffeomorphic to standard real projective spaces. First examples thereof were discovered by \textit{M. W. Hirsch} and \textit{J. W. Milnor} [Bull. Am. Math. Soc. 70, 372--377 (1964; Zbl 0201.25601)] as quotients of standard 5- and 6-spheres embedded in Milnor's exotic 7-spheres under certain free involutions. Similarly, \textit{P. Rajan} and \textit{F. Wilhelm} [Bull. Aust. Math. Soc. 94, No. 2, 304--315 (2016; Zbl 1364.53040)] detected some standard 13- and 14-spheres embedded in Shimada's exotic 15-spheres [\textit{N. Shimada}, Nagoya Math. J. 12, 59--69 (1957; Zbl 0145.20303)] with quotients that are homotopy equivalent to \(\mathbb{R}\mathrm{P}^{13}\) and \(\mathbb{R}\mathrm{P}^{14}\), respectively, and they also showed that some of the quotients in the 14-dimensional examples are not diffeomorphic to \(\mathbb{R}\mathrm{P}^{14}\), i.\,e.\ they are fake projective spaces.
The first main result of the publication under review is that some of the 13-dimensional homotopy projective spaces \(P^{13}\) found by Rajan and Wilhelm are fake projective spaces as well.
Further main results are concerned with invariant metrics of non-negative sectional curvature on fake projective spaces. It is known that the 5-dimensional Hirsch-Milnor fake projective spaces \(P^5\) admit cohomogeneity one actions by \(\mathsf{SO}(2)\times \mathsf{SO}(3)\) [\textit{M. W. Davis}, Am. J. Math. 104, 59--90 (1982; Zbl 0509.57029)] and that all of the \(P^5\)'s carry \(\mathsf{SO}(2)\times \mathsf{SO}(3)\)-invariant metrics of non-negative sectional curvature [\textit{K. Grove} and \textit{W. Ziller}, Ann. Math. (2) 152, No. 1, 331--367 (2000; Zbl 0991.53016), p.334]. Similarly, it is shown here that all Rajan-Wilhelm \(P^{13}\)'s admit cohomogeneity one actions by \(\mathsf{SO}(2)\times \mathsf{G}_2\), but in contrast to the 5-dimensional case the authors now prove the following alternative fact: None of the \(P^{13}\)'s supports an \(\mathsf{SO}(2)\times \mathsf{G}_2\)-invariant metric of non-negative sectional curvature.
In fact, it is shown that the 5-dimensional case is rather special in this regard: If a homotopy sphere admits a non-negatively curved metric that is invariant under a cohomogeneity one action, then the sphere is a standard sphere \(\mathbb{S}^n\) and either the action is linear, i.\,e.\ a sub-action of the standard action of \(\mathsf{SO}(n+1)\), or \(n=5\) and the action is nonlinear by \(\mathsf{SO}(2)\times \mathsf{SO}(3)\).
Moreover, the authors show that the 13-dimensional Rajan-Wilhelm fake projective spaces are \(\mathsf{SO}(2)\times \mathsf{G}_2\)-equivariantly diffeomorphic to quotients of Brieskorn varieties by involutions. This reformulation is the key technical resource in the proofs of the main results.Decomposition of some Witten-Reshetikhin-Turaev representations into irreducible factorshttps://www.zbmath.org/1472.570312021-11-25T18:46:10.358925Z"Korinman, Julien"https://www.zbmath.org/authors/?q=ai:korinman.julienFollowing a proposal of Witten [\textit{E. Witten}, Commun. Math. Phys. 121, No. 3, 351--399 (1989; Zbl 0667.57005)], Reshetikhin and Turaev defined Topological Quantum Field Theories from representations of quantum groups [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)]. Following the skein-theoretic approach developed in [\textit{W. B. R. Lickorish}, Pac. J. Math. 149, No. 2, 337--347 (1991; Zbl 0728.57011); \textit{C. Blanchet} et al., Topology 34, No. 4, 883--927 (1995; Zbl 0887.57009)], the paper under review studies the associated \(\mathrm{SU}(2)\)-representations of a central extension of the mapping class group of a closed surface. For a genus-\(2\) surface, these representations are decomposed into irreducible factors. Some partial results are given in higher genus.Monopole Floer homology and the spectral geometry of three-manifoldshttps://www.zbmath.org/1472.570322021-11-25T18:46:10.358925Z"Lin, Francesco"https://www.zbmath.org/authors/?q=ai:lin.francescoThere is a well-developed theory that examines the spectrum of the Laplace operator on manifolds. Less is known about the spectrum of the Laplace operator on forms. This paper uses Seiberg-Witten theory to derive an upper bound on the first eigenvalue of the Hodge Laplacian on co-exact \(1\)-forms on a wide class of \(3\)-manifolds. This result improves prior results that began with the Seiberg-Witten proof of the adjunction inequality. After establishing this upper bound, it is applied to give a new proof of an inequality first established for hyperbolic manifolds by \textit{J. F. Brock} and \textit{N. M. Dunfield} [Invent. Math. 210, No. 2, 531--558 (2017; Zbl 1379.57023)]. This is an example of a result that follows easily once one applies one key idea. In this case after establishing the standard inequality that implies compactness results for the Seiberg-Witten moduli via a Weitzenböck formula, Lin applies the Bochner formula to the form represented by the quadratic term in the first Seiberg-Witten formula. The result is a very clean and clear proof of this interesting result.Examples of contact mapping classes of infinite order in all dimensionshttps://www.zbmath.org/1472.570332021-11-25T18:46:10.358925Z"Gironella, Fabio"https://www.zbmath.org/authors/?q=ai:gironella.fabioLet \(V\) be a smooth \((2n+1)\)-manifold and \(\xi\) a contact structure on \(V\), that is, \(\xi\) is a hyperplane distribution on \(V\) satisfying a condition of complete non-integrability. In this paper the author studies the topology of the space of contactomorphisms \(\mathrm{Diff}(V,\xi)\) of the contact manifold \((V,\xi)\), comparing it with the one of the space of diffeomorphisms \(\mathrm{Diff}(V)\) of the manifold \(V\). It is known that the space \(\mathrm{Cont}(V)\) of contact structures on \(V\) plays an important role in the study of the relations between \(\mathrm{Diff}(V,\xi)\) and \(\mathrm{Diff}(V)\). Indeed, the map \(\mathrm{Diff}(V)\to\mathrm{Cont}(V)\), defined by \(\phi \mapsto \phi_*(\xi)\), is a locally-trivial fibration with fiber \(\mathrm{Diff}(V,\xi)\). This fibration induces a long exact sequence of homotopy groups \(\ldots \rightarrow \pi_{k+1}\left(\mathrm{Cont}(V)\right)\rightarrow \pi_k\left(\mathrm{Diff}(V,\xi)\right) \xrightarrow{j_*} \pi_k\left(\mathrm{Diff}(V)\right)\rightarrow \pi_k\left(\mathrm{Cont}(V)\right)\rightarrow\ldots \) where \(j_*:\pi_k\left(\mathrm{Diff}(V,\xi)\right)\rightarrow\pi_k\left(\mathrm{Diff}(V)\right)\) is the map induced on the homotopy groups by the natural inclusion \(j:\mathrm{Diff}(V,\xi)\rightarrow\mathrm{Diff}(V)\).
This paper focuses on the problem of the existence of infinite cyclic subgroups in \(\ker(j_*\vert_{\pi_{0}})\). The only known example of such a phenomenon is found in [\textit{R. E. Gompf}, Ann. Math. (2) 148, No. 2, 619--693 (1998; Zbl 0919.57012)] and [\textit{F. Ding} and \textit{H. Geiges}, Compos. Math. 146, No. 4, 1096--1112 (2010; Zbl 1209.57021)]. More precisely, Gompf argues that \(S^{2}\times S^1\), equipped with its unique (up to isotopy) tight contact structure \(\xi_{std}\), has a contact mapping class of infinite order. Then, starting from Gompf's remark, Ding and Geiges prove that \(\ker(j_*\vert_{\pi_{0}})\) and \(\pi_1(\mathrm{Con}(S^{2}\times S^1),\xi_{std})\) are actually both isomorphic to \(\mathbb{Z}\).
In this paper the author gives examples of tight high dimensional contact manifolds admitting a contactomorphism whose powers are all smoothly isotopic but not contact-isotopic to the identity. This is a generalization of an observation in dimension 3 by Gompf, also reused by Ding and Geiges.On compact abelian Lie groups of homeomorphisms of \(\mathbb{R}^m\)https://www.zbmath.org/1472.570342021-11-25T18:46:10.358925Z"Ben Rejeb, Khadija"https://www.zbmath.org/authors/?q=ai:ben-rejeb.khadijaThe continuous actions of compact abelian finite and Lie groups \(G\) on \(\mathbb{R}^m\) are investigated. It is known that in general such groups are not necessarily conjugate to subgroups of \(O(m)\). Here some special case is considered. Let \(S = S(K_1) \times \dots \times S(K_q)\), where \(K_i = \mathbb{R}\) or \(\mathbb{C}\) and \(S(K_i) = \{x \in K_i : |x| = 1 \}\) for \(1 \le i \le q\). These groups act naturally by homeomorphisms on \(\mathbb{R}^m = K_1 \oplus \dots \oplus K_q\). Let \(G\) be a compact Lie group of homeomorphisms of \(\mathbb{R}^m\). Is is shown that such \(G\) is contained in \(S\) if and only if every element of \(G\) centralizes \(S\). Corollary: \(G\) is conjugate to some subgroup of \(S\) if and only if for some homeomorphism \(\alpha\) of \(\mathbb{R}^m\) every element of \(\alpha G \alpha ^{-1}\) centralizes \(S\). Two examples showing the importance of the condition ``centralizes \(S\)'' are given.On cohomogeneity one linear actions on pseudo-Euclidean space \(\mathbb{R}^{p , q} \)https://www.zbmath.org/1472.570352021-11-25T18:46:10.358925Z"Ahmadi, P."https://www.zbmath.org/authors/?q=ai:ahmadi.parviz"Safari, S."https://www.zbmath.org/authors/?q=ai:safari.salimIn this article, the authors study the natural actions of some (noncompact) subgroups of \( SO(p,q) \) (matrices preserving the quadratic form of signature \( p,q \ (p\geq q)\)) on \( \mathbb{R}^{p+q} \). The case \( q=1 \) has been investigated in [\textit{J. Berndt} et al., Monatsh. Math. 184, No. 2, 185--200 (2017; Zbl 1379.53091)]. In this article, similar results are obtained for \( q>1 \). All orbits of this cohomogeniety one action are found. A decomposition of the subgroup \( Q \) leaving a certain subspace invariant is determined. Restricted actions to a certain class of subgroups of \( Q \) are studied in detail. The authors prove that the actions of these subgroups are also of cohomogeniety one and all their orbits are determined. It is shown that, when \( p>q+1\geq2 \), the orbits outside a certain subspace (\( \mathbb{W} \) of dimension \( p \)) are independent of the acting group (in this class) but the orbits of points in \( \mathbb{W} \) depend on the acting group as in [loc. cit.].Nielsen spectrum of maps on infra-solvmanifolds modeled on \(\operatorname{Sol}_0^{\kern2pt 4}\)https://www.zbmath.org/1472.570362021-11-25T18:46:10.358925Z"Lee, Jong Bum"https://www.zbmath.org/authors/?q=ai:lee.jong-bumThe author considers Nielsen theory of an infra-solvmanifold \(M\) modeled on the 4-dimensional solvable Lie group \(\text{Sol}_0^4\). Given a self-map \(f\) on \(M\), each of the classical numerical Nielsen theory invariants is computed in terms of an algebraic parameter \(k\) which is easily deduced from the structure of \(f\). This \(k\) can take on any integer value.
The author shows that the Reidemeister, Lefschetz, and Nielsen numbers of \(f\) take the following form:
\begin{align*}
R(f) &= \sigma(1-k) \\
L(f) &= 1-k \\
N(f) &= |1-k|
\end{align*}
where \(\sigma(n)=\infty\) when \(n=0\) and \(\sigma(n)=|n|\) otherwise.
Immediately we can deduce the ``spectrum'' of these invariants, that is, the set of all possible values that the invariant can attain for any selfmap of an infra-solvmanifold modeled on \(\text{Sol}_0^4\). The spectrum of \(R(f)\) is \(\mathbb N \cup \{\infty\}\) (where \(\mathbb N\) does not include 0), the spectrum of \(L(f)\) is \(\mathbb Z\), and the spectrum of \(N(f)\) is \(\mathbb N \cup \{0\}\).Foliated manifolds, algebraic \(K\)-theory, and a secondary invarianthttps://www.zbmath.org/1472.580162021-11-25T18:46:10.358925Z"Bunke, Ulrich"https://www.zbmath.org/authors/?q=ai:bunke.ulrichIn this paper, Bunke introduces a numerical invariant (taking values in \({\mathbb{C}}/{\mathbb{Z}}\)) associated to foliations over odd-dimensional closed spin manifolds by means of a certain transgression of the \(\hat{A}\) form. The paper is carefully written and contains a wealth of examples making it a very pleasurable read.
The foliation \(\mathcal{F}\), defined as an involutive \(\mathbb{C}\)-vector subbundle in the complexification of the tangent bundle, is assumed to be stably trivial and equipped with a stable framing \(s\). Additionally, one fixes a complex vector bundle \(V\) over the base manifold \(M\), together with a partial connection in the direction of the foliation (a partial connection means the restriction of a connection to those covariant derivatives in the direction of the foliation). The partial connection \(\nabla^I\) on \(V\) is assumed to be flat. The normal bundle to the foliation \(T_{\mathbb{C}}M/{\mathcal{F}}\) is equipped with a natural such flat partial connection, given by the Lie bracket.
Under these hypothesis, the invariant \(\rho(M,\mathcal{F},\nabla^I,s)\in {\mathbb{C}}/{\mathbb{Z}}\) is constructed by fixing some additional geometric data: an extension of the flat partial connection to an actual connection on \(V\), a Riemannian metric on \(M\), and also an extension of the natural flat partial connection on the transverse bundle to the foliation to an actual connection over \(M\). However, if the codimension of the foliation is small enough in the sense that \(2\mathrm{codim}(\mathcal{F})<\dim(M)\), then the invariant is shown to be independent of the above additional data.
The definition of invariant involves the so-called Umkehr map in differential \(K\)-theory. In particular cases, it reduces to known invariants like the (reduced) eta invariant of twisted Dirac operators on spin manifolds, the Godbillon-Vey invariant of a foliation of codimension \(1\), or Adams' \(e\)-invariant. In the last sections of the paper, the invariant is linked to a regulator map in algebraic \(K\)-theory.Sufficient descent Riemannian conjugate gradient methodshttps://www.zbmath.org/1472.650722021-11-25T18:46:10.358925Z"Sakai, Hiroyuki"https://www.zbmath.org/authors/?q=ai:sakai.hiroyuki"Iiduka, Hideaki"https://www.zbmath.org/authors/?q=ai:iiduka.hideakiSummary: This paper considers sufficient descent Riemannian conjugate gradient methods with line search algorithms. We propose two kinds of sufficient descent nonlinear conjugate gradient method and prove that these methods satisfy the sufficient descent condition on Riemannian manifolds. One is a hybrid method combining a Fletcher-Reeves-type method with a Polak-Ribière-Polyak-type method, and the other is a Hager-Zhang-type method, both of which are generalizations of those used in Euclidean space. Moreover, we prove that the hybrid method has a global convergence property under the strong Wolfe conditions and the Hager-Zhang-type method has the sufficient descent property regardless of whether a line search is used or not. Further, we review two kinds of line search algorithm on Riemannian manifolds and numerically compare our generalized methods by solving several Riemannian optimization problems. The results show that the performance of the proposed hybrid methods greatly depends on the type of line search used. Meanwhile, the Hager-Zhang-type method has the fast convergence property regardless of the type of line search used.Gauge fields and quantum entanglementhttps://www.zbmath.org/1472.811682021-11-25T18:46:10.358925Z"Mielczarek, Jakub"https://www.zbmath.org/authors/?q=ai:mielczarek.jakub"Trześniewski, Tomasz"https://www.zbmath.org/authors/?q=ai:trzesniewski.tomaszSummary: The purpose of this letter is to explore the relation between gauge fields, which are at the base of our understanding of fundamental interactions, and the quantum entanglement. To this end, we investigate the case of SU(2) gauge fields. It is first argued that holonomies of the SU(2) gauge fields are naturally associated with maximally entangled two-particle states. Then, we provide some evidence that the notion of such gauge fields can be deduced from the transformation properties of maximally entangled two-particle states. This new insight unveils a possible relation between gauge fields and spin systems, as well as contributes to understanding of the relation between tensor networks (such as MERA) and spin network states considered in loop quantum gravity. In consequence, our results turn out to be relevant in the context of the emerging Entanglement/Gravity duality.