Recent zbMATH articles in MSC 57https://www.zbmath.org/atom/cc/572021-02-12T15:23:00+00:00WerkzeugInvariants and TQFT's for cut cellular surfaces from finite 2-groups.https://www.zbmath.org/1452.570252021-02-12T15:23:00+00:00"Bragança, Diogo"https://www.zbmath.org/authors/?q=ai:braganca.diogo"Picken, Roger"https://www.zbmath.org/authors/?q=ai:picken.roger-fSummary: In this brief sequel to a previous article \textit{D. Bragança} and \textit{R. Picken} [Bol. Soc. Port. Mat. 74, 17--44 (2016; Zbl 07301091)], we recall the notion of a cut cellular surface (CCS), being a surface with boundary, which is cut in a specified way to be represented in the plane, and is composed of 0-, 1- and 2-cells. We obtain invariants of CCS's under Pachner-like moves on the cellular structure, by counting colorings of the 1- and 2-cells with elements of a finite 2-group, subject to a ``fake flatness'' condition for each 2-cell. These invariants, which extend Yetter's invariants to this class of surfaces, are also described in a TQFT setting. A result from the previous article concerning the commuting fraction of a group is generalized to the 2-group context.Profinite rigidity, fibering, and the figure-eight knot.https://www.zbmath.org/1452.570132021-02-12T15:23:00+00:00"Bridson, Martin R."https://www.zbmath.org/authors/?q=ai:bridson.martin-r"Reid, Alan W."https://www.zbmath.org/authors/?q=ai:reid.alan-wThe paper has the same motivation and some overlapping results (by different methods) with the second paper by \textit{M. Boileau} and \textit{S. Friedl} in the present volume [Ann. Math. Stud. 205, 21--44 (2020; Zbl 1452.57012)]; it is shown e.g. that the complement of the figure-8 knot is determined among compact connected 3-manifolds by its profinite completion; also, the authors note that in the meantime \textit{H. Wilton} and \textit{P. Zalesskii} proved a profinite recognition theorem for finite volume hyperbolic 3-manifolds [Compos. Math. 155, No. 2, 246--259 (2019; Zbl 1436.57020)].
For the entire collection see [Zbl 1437.55002].
Reviewer: Bruno Zimmermann (Trieste)The profinite completion of 3-manifold groups, fiberedness and the Thurston norm.https://www.zbmath.org/1452.570122021-02-12T15:23:00+00:00"Boileau, Michel"https://www.zbmath.org/authors/?q=ai:boileau.michel"Friedl, Stefan"https://www.zbmath.org/authors/?q=ai:friedl.stefanThe paper centers around the question of what properties of a compact 3-manifold are determined by the profinite completion of its fundamental group (or the collection of its finite quotients); it is shown e.g. that two knots in \(S^3\) whose groups have isomorphic profinite completions have the same genus, and are both either fibered or not fibered, and that torus knots and the figure-8 knot are determined among all knots by the profinite completions of their groups.
For the entire collection see [Zbl 1437.55002].
Reviewer: Bruno Zimmermann (Trieste)A Bieberbach theorem for crystallographic group extensions.https://www.zbmath.org/1452.200502021-02-12T15:23:00+00:00"Ratcliffe, John G."https://www.zbmath.org/authors/?q=ai:ratcliffe.john-g"Tschantz, Steven T."https://www.zbmath.org/authors/?q=ai:tschantz.steven-tSummary: In this paper, we prove that for each dimension \(n\), there are only finitely many isomorphism classes of pairs of groups \((\Gamma,N)\) such that \(\Gamma\) is an \(n\)-dimensional crystallographic group and \(N\) is a normal subgroup of \(\Gamma\) such that \(\Gamma/N\) is a crystallographic group. This result is equivalent to the statement that for each dimension \(n\) there are only finitely many affine equivalence classes of geometric orbifold fibrations of compact, connected and flat \(n\)-orbifolds.Real algebraic links in \(S^3\) and braid group actions on the set of \(n\)-adic integers.https://www.zbmath.org/1452.570042021-02-12T15:23:00+00:00"Bode, Benjamin"https://www.zbmath.org/authors/?q=ai:bode.benjaminThe goal of the paper under review is to construct braid group actions on the set of the \(n\)-adic integers \(\mathbb{Z}_n\). Such study leads to connections between different aspects of the study of braid groups, linking group theoretic properties with the topology of certain configuration spaces and subsets of the space of complex polynomials. Let \(C_n\) be the space of monic complex polynomials of degree \(n\), and \[V_n:=\{(v_1, v_2, \ldots , v_{n-1})\in (\mathbb{C}\backslash\{0\})^{n-1}: v_i\ne v_j \text{ if } i\ne j \}/S_{n-1}.\]
The author proves:
Theorem 1.2. There is a tower of covering spaces
\[\cdots \to Z_{n}^{i+1} \to Z_n^i \to \cdots \to Z_n^2 \to Z_n^1=Z_n\to V_n\simeq D_n\stackrel{p}\to C_n,\]
\noindent where \(p\) is a covering map of degree \(n\), all other arrows are covering maps of degree \(n^{n-1}\) and \(\simeq\) denotes homotopy equivalence.
The fiber over a point \(v\in V_n\) is the set of \(n^{n-1}\)-adic integers \(\mathbb{Z}_{n^{n-1}}\). This provides an action of \(\pi_1(C_n)\), which is the braid group, on the set \(\mathbb{Z}/n\mathbb{Z}\times \mathbb{Z}_{n^{n-1}}\), via monodromy. Write this action as \(\phi_n(\ .\ , B):\mathbb{Z}_n \to \mathbb{Z}_n, \ B\in \mathbb{B}_n\).
With very similar considerations, another action \(\psi_n\) of the braid group on \(\mathbb{Z}_{n^n}\cong \mathbb{Z}_n\) is constructed. The author shows:
Theorem 1.3. For both of the constructed actions \(\phi_n\) and \(\psi_n\), the following is true.
\begin{itemize}
\item[(i)] They preserve the metrics on \(\mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}_{n^{n-1}}\) and \( \mathbb{Z}_{n^{n}}\), respectively. Therefore, they yield continuous braid group actions on \(\mathbb{Z}_n\).
\item[(ii)] They correspond to sequences of homomorphisms \(\mathcal{B}_n \to S_{n\times (n^{n-1})^j}\) and \(\mathcal{B}_n \to S_{ (n^{n})^j}\), respectively. For \(\phi_n\), the resulting action on \(n\times(n^{n-1})^j\) points is transitive for all \(n\) and \(j\).
\item[(iii)] The kernels \(N_j\) and \(H_j\) of the homomorphisms \(\mathcal{B}_n \to S_{n\times (n^{n-1})^j}\) and \(\mathcal{B}_n \to S_{ (n^{n})^j}\), respectively, form descending series of normal subgroups of the braid group that do not stabilize.
\end{itemize}
The paper contains a quite complete and useful exposition of the background which is used in the paper concerning the \(n\)-adic integers, polynomials, braids and covering maps.
It is explained how this material can be used to improve invariants of braids or conjugacy classes. Also the sequences of normal subgroups that are given by the kernels of the homomorphisms given in the statement of Theorem 1.3 (ii) are studied, and it is shown that they do not stabilize. Calculation with the action \(\psi_n\) is used to show an infinite family of braids close to real algebraic links. The author obtains the following result:
Theorem 1.4. Let \(\epsilon \in \{\pm 1\}\) and let \(B=\Pi_{j+1}^{\ell}\omega_{i_j}^{\epsilon}\) with \(i_j \in \{1,2,\ldots,5\}\) be a \(3\)-strand braid with \[\omega_1=\sigma_2, \ \ \ \ \omega_2=\sigma_1^2, \ \ \ \ \omega_3=(\sigma_1\sigma_2\sigma_1)^2,\] \[\omega_4=(\sigma_2\sigma_1\sigma_2^{-1}\sigma_1\sigma_2)^2, \ \ \ \ \omega_5=\sigma_2^{-1}\sigma_1\sigma_2\sigma_2\sigma_1,\] and such that there is a \(j\) such that \(i_j=3\) or such that there is only one residue class \(k\) mod \(3\) such that \(i_j\ne k\) for all \(j=1,2,\ldots, \ell\). Then, the closure of \(B^2\) is real algebraic.
Reviewer: Daciberg Lima Gonçalves (São Paulo)Twistor geometry and gauge fields.https://www.zbmath.org/1452.811512021-02-12T15:23:00+00:00"Sergeev, Armen"https://www.zbmath.org/authors/?q=ai:sergeev.armen-glebovichSummary: In our course we have presented the basics of twistor theory and its applications to the solution of Yang-Mills duality equations. The first part describes the twistor correspondence between geometric objects in Minkowski space and their counterparts in twistor space. In the second part we apply twistor theory to the study of Yang-Mills duality equations on \(\mathbb{R}^4\). We include a list of references for further study.
For the entire collection see [Zbl 1433.53003].Virtual rational tangles.https://www.zbmath.org/1452.570092021-02-12T15:23:00+00:00"Mellor, Blake"https://www.zbmath.org/authors/?q=ai:mellor.blake"Nevin, Sean"https://www.zbmath.org/authors/?q=ai:nevin.seanCertifying the Thurston norm via SL\((2,\mathbb{C})\)-twisted homology.https://www.zbmath.org/1452.570112021-02-12T15:23:00+00:00"Agol, Ian"https://www.zbmath.org/authors/?q=ai:agol.ian"Dunfield, Nathan M."https://www.zbmath.org/authors/?q=ai:dunfield.nathan-mFor a large class of hyperbolic knots including fibered knots, the conjecture from [\textit{N. M. Dunfield} et al., Exp. Math. 21, No. 4, 329--352 (2012; Zbl 1266.57008)] is proved that the hyperbolic torsion polynomial (a refined version of the twisted Alexander polynomial) determines the Seifert genus of the knot.
For the entire collection see [Zbl 1437.55002].
Reviewer: Bruno Zimmermann (Trieste)Sufficient condition for tangential transversality.https://www.zbmath.org/1452.490142021-02-12T15:23:00+00:00"Apostolov, Stoyan R."https://www.zbmath.org/authors/?q=ai:apostolov.stoyan-r"Krastanov, Mikhail I."https://www.zbmath.org/authors/?q=ai:krastanov.mikhail-ivanov"Ribarska, Nadezhda K."https://www.zbmath.org/authors/?q=ai:ribarska.nadezhda-kTransversality coming from mathematical analysis and differential topology is also an extremely natural and convenient concept in some parts of variational analysis (see [\textit{A. D. Ioffe}, J. Optim. Theory Appl. 174, No. 2, 343--366 (2017; Zbl 1382.49014)]). The extension to subtransversality [\textit{D. Drusvyatskiy} et al., Found. Comput. Math. 15, No. 6, 1637--1651 (2015; Zbl 1338.49057)] for two closed sets \(A,B\) at \(x_0 \in A\cap B\), given by \(d(x,A\cap B)\le K (d(x,A)+d(x,B))\) for some \(K>0\), some neighborhood \(U\) of \(x_0\) and all \(x \in U\), is very useful for deriving necessary optimality conditions of the Pontryagin maximum principle. In the article, suffficient conditions of tangential transversality introduced by \textit{M. Bivas} et al. [J. Math. Anal. Appl. 481, No. 1, Article ID 123445, 21 p. (2020; Zbl 1432.49017)] are investigated and applied to abstract optimal control in B-spaces. Tangential transversality is sufficient for subtransversality [Bivas et al., loc. cit.]. The well-known sufficient condition for tangential transversality using compactly epi-Lipschitz (massive) sets is weakened to a symmetric condition with respect to the sets \(A,B\) taking uniform tangent sets of corresponding tangent cones for proving tangential transversality of \(A,B\) at \(x_0\) (Th. 3.1 and Th. 3.2). The result yields an abstract version of the well-known Aubin condition from [\textit{F. Clarke}, Necessary conditions in dynamic optimization. Providence, RI: American Mathematical Society (AMS) (2005; Zbl 1093.49017)] which is then applied to an abstract optimal control problem in Banach spaces. They introduce as specialization of the above symmetric condition so-called jointly massive sets \(A,B\) -- splitting the massiveness between \(A\) and \(B\) -- which ensure tangential transversality too. The proofs are given in detail for new results.
For better understanding the ideas behind this paper, study first [Bivas et al., loc. cit.].
Reviewer: Armin Hoffmann (Ilmenau)Coarse homology of leaves of foliations.https://www.zbmath.org/1452.570232021-02-12T15:23:00+00:00"Schmidt, Robert"https://www.zbmath.org/authors/?q=ai:schmidt.robert-l|schmidt.robert-c|schmidt.robert-wThis paper computes coarse homology of Riemannian manifolds which arise as leaves of foliations.
The author extends a result by \textit{P. A. Schweitzer} [Ann. Inst. Fourier 61, No. 4, 1599--1631 (2011; Zbl 1241.57036)]. He shows that there exist examples of leaves in compact manifolds with infinite rank coarse homology in degree 1. For this purpose spaces with infinitely many ends are constructed.
Conversely the paper presents examples for Riemannian manifolds with trivial coarse homology which do not arise as leaves of codimension one \(C^{2,0}\)-foliations of compact manifolds.
Reviewer: Elisa Hartmann (Göttingen)Twisted torus knots \(T(p,q,p-kq,-1)\) which are torus knots.https://www.zbmath.org/1452.570062021-02-12T15:23:00+00:00"Lee, Sangyop"https://www.zbmath.org/authors/?q=ai:lee.sangyopTau invariants for balanced spatial graphs.https://www.zbmath.org/1452.570212021-02-12T15:23:00+00:00"Vance, Katherine"https://www.zbmath.org/authors/?q=ai:vance.katherineSemimeander crossing number of knots and related invariants.https://www.zbmath.org/1452.570032021-02-12T15:23:00+00:00"Belousov, Yu."https://www.zbmath.org/authors/?q=ai:belousov.yury|belousov.yu-iSummary: The minimum number of crossings among all of the diagrams of a knot \(K\) composed of at most \(k\) smooth simple arcs is called the \(k\)-arc crossing number of \(K\). This number is denoted by \(\operatorname{cr}_k(K)\). The 2-arc crossing number is also called the semimeander crossing number. The article studies connections of the \(k\)-arc crossing numbers with the classical crossing number \(\operatorname{cr}(K)\) of \(K\). It is proved that for each knot \(K\), the following inequalities are fulfilled: \(\operatorname{cr}_2(K)\le \sqrt[4]{6}^{\operatorname{cr}(K)}\) and \(\operatorname{cr}_k (K)\le \operatorname{cr}_{k+1}(K)+2\frac{(\operatorname{cr}_{k+1}(K))^2}{(k+1)^2}\).Lower central and derived series of semi-direct products, and applications to surface braid groups.https://www.zbmath.org/1452.200332021-02-12T15:23:00+00:00"Guaschi, John"https://www.zbmath.org/authors/?q=ai:guaschi.john"de Miranda e Pereiro, Carolina"https://www.zbmath.org/authors/?q=ai:de-miranda-e-pereiro.carolinaIn the first part of the present paper, the authors give a general description of lower central series and derived series an arbitrary semi-direct product. In the second part of the paper, they study these series for the full braid group \(B_n(M)\) and pure braid group \(P_n(M)\) of a compact surface \(M\), orientable or non-orientable, the aim being to determine the values of \(n\) for which \(B_n(M)\) and \(P_n(M)\) are residually nilpotent or residually soluble.
They first solve this problem in the case where \(M\) is the 2-torus \(\mathbb{T}\). The authors prove (Theorem 1.2) that the braid group \(B_n(\mathbb{T})\) is residually soluble if and only if \(n \leq 4\).
For the braid group of the Klein bottle is proved that \(P_n(\mathbb{K})\) is residually nilpotent for all \(n \geq 1\) (see Theorem 1.3).
In Theorem 1.4 is proved that if \(M\) is a compact non-orientable surface of genus \(g \geq 1\) without boundary, then \(B_n(M)\) is residually nilpotent if and only if \(n \leq 2\), and is residually soluble if and only if \(n \leq 4\).
Reviewer: Valeriy Bardakov (Novosibirsk)Divergence and parallelism of cylindrical stretch lines.https://www.zbmath.org/1452.300232021-02-12T15:23:00+00:00"Théret, Guillaume"https://www.zbmath.org/authors/?q=ai:theret.guillaumeSummary: A cylindrical stretch line is a stretch line, in the sense of Thurston, whose horocyclic lamination is a weighted multicurve. In this paper, we show that two correctly parameterized cylindrical lines are parallel if and only if these lines converge towards the same point in Thurston's boundary of Teichmüller space.Essential surfaces in graph pairs.https://www.zbmath.org/1452.200382021-02-12T15:23:00+00:00"Wilton, Henry"https://www.zbmath.org/authors/?q=ai:wilton.henrySummary: A well-known question of Gromov asks whether every one-ended hyperbolic group \( \Gamma \) has a surface subgroup. We give a positive answer when \( \Gamma \) is the fundamental group of a graph of free groups with cyclic edge groups. As a result, Gromov's question is reduced (modulo a technical assumption on 2-torsion) to the case when \( \Gamma \) is rigid. We also find surface subgroups in limit groups. It follows that a limit group with the same profinite completion as a free group must in fact be free, which answers a question of Remeslennikov in this case.Knot concordances in \(S^1\times S^2\) and exotic smooth \(4\)-manifolds.https://www.zbmath.org/1452.570162021-02-12T15:23:00+00:00"Akbulut, Selman"https://www.zbmath.org/authors/?q=ai:akbulut.selman"Yildiz, Eylem Zeliha"https://www.zbmath.org/authors/?q=ai:yildiz.eylem-zelihaIn this paper, the authors first prove that any knot \(K\) in \(S^1 \times S^2\), which is freely homotopic to \(S^1 \times \textrm{pt}\), is invertibly concordant to \(S^1 \times \textrm{pt}\). The knots \(K \subset S^1 \times S^2\) with hyperbolic complements and trivial symmetry group are of special interest here, because they can be used to generate absolutely exotic (the exotic structure is not relative to a particular parameterization of the boundary) compact \(4\)-manifolds by the recipe given by \textit{S. Akbulut} and \textit{D. Ruberman} [Comment. Math. Helv. 91, No. 1, 1--19 (2016; Zbl 1339.57003)].
In the paper under review, the authors build absolutely exotic manifold pairs by this construction, and show that this construction keeps the Stein property of the \(4\)-manifolds they start with. They also establish the existence of an absolutely exotic contractible Stein manifold pair, and an absolutely exotic homotopy \(S^1 \times B^3\) Stein manifold pair.
Reviewer: Mehmetcik Pamuk (Ankara)Structure of symmetry group of some composite links and some applications.https://www.zbmath.org/1452.570072021-02-12T15:23:00+00:00"Liu, Yang"https://www.zbmath.org/authors/?q=ai:liu.yang.22|liu.yang.23|liu.yang.21|liu.yang.13|liu.yang.3|liu.yang.4|liu.yang.9|liu.yang.10|liu.yang|liu.yang.2|liu.yang.20|liu.yang.18|liu.yang.12|liu.yang.19|liu.yang.6|liu.yang.14|liu.yang.8|liu.yang.16|liu.yang.11|liu.yang.15|liu.yang.5|liu.yang.7|liu.yang.1|liu.yang.17This paper considers the symmetry groups of topological links, primarily in three space. The link \(2^{2}_{1}m\#2^{2}_{1}\) is used as an example for the methods of the paper.
Reviewer: Jonathan Hodgson (Swarthmore)Lens spaces which are realizable as closures of homology cobordisms over planar surfaces.https://www.zbmath.org/1452.570152021-02-12T15:23:00+00:00"Sekino, Nozomu"https://www.zbmath.org/authors/?q=ai:sekino.nozomuIn [Ill. J. Math. 62, No. 1--4, 99--111 (2018; Zbl 1409.57017)] \textit{Y. Nozaki} showed by solving algebraic equations that every lens space contains a genus-one homologically fibered knot. Like him, the author gives us a relation between an algebraic equation and the existence of a homologically fibered link of a given number of components whose Seifert surface is a planar surface in a lens space. Denote by \(L(p, q)\) a lens space and by \(\Sigma_{g, n}\) a surface of genus \(g\) having \(n\) boundary components; we first consider the following
Proposition 1.
\(L(p, q)\) has a realization as a closure of a homology cobordism over \(\Sigma_{0, n+1}\) for \(n \in \mathbb N\) if and only if there are integers \(a_h\), \(l_{i, j}\) and \(t_k\), where \(1 \leq h, i, j, k \leq n\) with \(i \leq j\), satisfying
\[
\det\begin{bmatrix} p &-qa_1&-qa_2& \dotsb &-qa_n\\ a_1&t_1&l_{1, 2}& \dotsb& l_{1,n}&\\a_2&l_{1, 2}&t_2& \dotsb &l_{2, n}\\ \vdots&\vdots & \vdots &\ddots &\vdots\\ a_n& l_{1, n}& l_{2, n}& \dotsb &t_n\end{bmatrix}=\pm1.
\]
Since any manifold that has a realization as a closure of a homology cobordism over \(\Sigma_{0, 1}\) is a homology \(3\)-sphere, none of the lens spaces are realizable as such. Further, if the determinant of the \(n\times n\) matrix above has a solution, then we may let \(a_{n+1} = l_{i, n+1}=0\) for all \(1 \leq i \leq n\) and \(t_{n+1}=1\) to obtain a solution for a matrix having a larger dimension than the one in the \(n\times n\) case. In other words, a lens space having a realization as a closure of a homology cobordism over \(\Sigma_{0, n+1}\) has one over \(\Sigma_{0, n+2}\). In this sense, the best way to think about the value \(n\) is to find its minimal number in the above matrix which still has a solution. In this article, the author answers this question as stated below.
Theorem 1.
\(L(p, q)\) has a realization as a closure of a homology cobordism over \(\Sigma_{0, 2}\) if and only if \(q\) or \(-q\) is a quadratic residue modulo \(p\).
Proof.
The proof of this theorem completely relies on the first proposition which is interestingly a straightforward idea. \(L(p, q)\) is realizable as a closure of a homology cobordism over \(\Sigma_{0, 2}\) is equivalent to the existence of integers \(a\) and \(t\) satisfying \(\det\begin{bmatrix} p &-qa\\ a& t\end{bmatrix}=\pm1\).
Hence, \(tp +qa^2 = \pm1\). Thus, \(p\) and \(a^2\) are relatively prime meaning \(p\) and \(a\) are coprime as well. Further, \(a\) satisfies \(qa^2 \equiv \pm 1 \pmod p\). As \(gcd\{p, a\}=1\), we obtain \(q \equiv \pm (a^{-1})^2 \pmod p\). Consequently, the condition for \(L(p, q)\) being realizable as a closure of a homology cobordism over \(\Sigma_{0, 2}\) is equivalent to \(q\) or \(-q\) becoming a quadratic residue modulo \(p\).
Theorem 2.
Every lens space has a realization as a closure of a homology cobordism over \(\Sigma_{0, 3}\).
Finally, I think that [\textit{H. Goda} and \textit{T. Sakasai}, Tokyo J. Math. 36, No. 1, 85--111 (2013; Zbl 1287.57022)] and [\textit{T. Sakasai}, Winter Braids Lect. Notes 3, Exp. No. 4, 25 p. (2016; Zbl 1422.57051)] are excellent references to understand related problems discussed in this paper.
Reviewer: Ryo Ohashi (Wilkes-Barre)Horo-flat surfaces along cuspidal edges in the hyperbolic space.https://www.zbmath.org/1452.530072021-02-12T15:23:00+00:00"Izumiya, Shyuichi"https://www.zbmath.org/authors/?q=ai:izumiya.shyuichi"Romero-Fuster, Maria Carmen"https://www.zbmath.org/authors/?q=ai:romero-fuster.maria-carmen"Saji, Kentaro"https://www.zbmath.org/authors/?q=ai:saji.kentaro"Takahashi, Masatomo"https://www.zbmath.org/authors/?q=ai:takahashi.masatomoSummary: There are two important classes of surfaces in the hyperbolic space. One consists of extrinsic flat surfaces, which is a notion analogous to developable surfaces in the Euclidean space. Another class consists of horo-flat surfaces, which are given by one-parameter families of horocycles. We use the Legendrian dualities between hyperbolic space, de Sitter space and the lightcone in the Lorentz-Minkowski 4-space in order to study the geometry of flat surfaces defined along the singular set of a cuspidal edge in the hyperbolic space. Such flat surfaces can be considered as flat approximations of the cuspidal edge. We investigate the geometric properties of a cuspidal edge in terms of the special properties of its flat approximations.Representations of the necklace braid group: topological and combinatorial approaches.https://www.zbmath.org/1452.200322021-02-12T15:23:00+00:00"Bullivant, Alex"https://www.zbmath.org/authors/?q=ai:bullivant.alex"Kimball, Andrew"https://www.zbmath.org/authors/?q=ai:kimball.andrew"Martin, Paul"https://www.zbmath.org/authors/?q=ai:martin.paul-purdon"Rowell, Eric C."https://www.zbmath.org/authors/?q=ai:rowell.eric-cThe necklace braid group (of rank \(n\)) is the motion group of the \(n\)-necklace which consists of \(n\) pairwise unlinked circles each linked to an auxiliary circle. Alternatively, the necklace group can be seen as an extension of the braid group by one extra generator (in the sense that it contains the braid group, and the quotient is generated by one element), namely, that corresponds to rotating the auxilary circle in the necklace. By the latter description, given a braid group representation, one can try to extend it to a representation of the necklace braid group by finding an appropriate action of the extra generator.
Constructed in the paper are so-called standard extensions, which exist for all irreducible braid group representations. The paper then discusses standard and non-standard extensions of various braid group representaions: the standard representation; the Burau representation; the Lawrence-Krammer-Bigelow representation; and some local representations, e.g., the representations coming from a braided fusion category, as well as a physics aspect of certain standard extensions.
The paper also describes a relation (a group morphism) between the necklace braid group and the loop braid group and compares some of the above results with the case for the loop braid group.
Reviewer: Hankyung Ko (Uppsala)On \(\mathbb{P} M\)-monoids and braid \(\mathbb{P} M\)-monoids.https://www.zbmath.org/1452.200612021-02-12T15:23:00+00:00"Miyatani, Toshinori"https://www.zbmath.org/authors/?q=ai:miyatani.toshinoriThe author defines two monoids analogous to the symmetric group \(S_n\) and the braid group \(B_n\), which are called \(\mathbb{P}M\) monoid and braid \(\mathbb{P}M\) monoid respectively. He shows that a \(\mathbb{P}M\) monoid has a presentation with generators and relators. Moreover he gives a solution to the word problem of the braid \(\mathbb{P}M\) monoid.
Reviewer: Stephan Rosebrock (Karlsruhe)A Fox-Milnor theorem for the Alexander polynomial of knotted 2-spheres in \(S^4\).https://www.zbmath.org/1452.570182021-02-12T15:23:00+00:00"Moussard, Delphine"https://www.zbmath.org/authors/?q=ai:moussard.delphine"Wagner, Emmanuel"https://www.zbmath.org/authors/?q=ai:wagner.emmanuelA 2-knot is the image of a smooth embedding of a 2-sphere into \(S^4\). A 2-knot is ribbon if it bounds a 3-ball with only special singularities called ribbon singularities. It is known that for any polynomial \(f(t) \in \mathbb{Z}[t^{\pm 1}]\) such that \(f(1)=1\), there is a ribbon 2-knot whose Alexander polynomial is \(f(t)\).
In this paper, the authors introduce the notion of an \(A\)-ribbon 2-knot, which is defined as a 2-knot bounding a 3-ball whose singularities consist only of ``annular ribbon (\(A\)-ribbon)'' singularities. In particular, ribbon 2-knots are \(A\)-ribbon. For \(A\)-ribbon 2-knots, the authors discuss the construction of Seifert hypersurfaces and compute Seifert matrices. And they define a condition called the linkings condition, and show the main result as follows. The Alexander polynomial of an \(A\)-ribbon 2-knot satisfying the linkings condition factorizes as \(f(t)f(t^{-1})\) for some polynomial \(f(t)\).
Reviewer: Inasa Nakamura (Kanazawa)Measuring similarity between curves on 2-manifolds via homotopy area.https://www.zbmath.org/1452.682442021-02-12T15:23:00+00:00"Chambers, Erin Wolf"https://www.zbmath.org/authors/?q=ai:chambers.erin-wolf"Wang, Yusu"https://www.zbmath.org/authors/?q=ai:wang.yusuSummary: Measuring the similarity of curves is a fundamental problem arising in many application fields. There has been considerable interest in several such measures, both in Euclidean space and in more general setting such as curves on Riemannian surfaces or curves in the plane minus a set of obstacles. However, so far, efficiently computable similarity measures for curves on general surfaces remain elusive. This paper aims at developing a natural curve similarity measure that can be easily extended and computed for curves on general orientable 2-manifolds. Specifically, we measure similarity between homotopic curves based on how hard it is to deform one curve into the other one continuously, and define this ``hardness'' as the minimum possible surface area swept by a homotopy between the curves. We consider cases where curves are embedded in the plane or on a triangulated orientable surface with genus $g$, and we present efficient algorithms (which are either quadratic or near linear time, depending on the setting) for both cases. The results are also extended to comparing simple cycles (simple closed curves) in the plane or on a sphere, although the algorithms become less efficient by a factor of $n$. We also discuss the case of cycles on surfaces, which remains open.Nielsen theory on infra-nilmanifolds modeled on the group of uni-triangular matrices.https://www.zbmath.org/1452.570262021-02-12T15:23:00+00:00"Choi, Younggi"https://www.zbmath.org/authors/?q=ai:choi.younggi"Lee, Jong Bum"https://www.zbmath.org/authors/?q=ai:lee.jong-bum"Lee, Kyung bai"https://www.zbmath.org/authors/?q=ai:lee.kyung-baiLet \(\text{Nil}_m\) be the group of uni-triangular real matrices of order \(m\).
There are three special cases here: \(m=2\) (\(\text{Nil}_2\) is isomorphic to \(\mathbb{R}^2\)), \(m=3\) (\(\text{Nil}_3\) is the Heisenberg group) and \(m \ge 4\). Then let \(\Gamma_m = \text{Nil}_m(\mathbb Z)\) be the discrete subgroup of integral points in \(\text{Nil}_m\) and \(\Pi\) be some Bieberbach group for \(\text{Nil}_m\) having \(\Gamma_m\) as its nil-radical. In this article the sets of all possible values (the spectra) of the Lefschetz, the Nielsen and the Reidemeister coincidence numbers for self-maps \(f\) of а infra-nilmanifolds \(M=\text{Nil}_m/\Pi\) are calculated (it is known that every such a map \(f\) is homotopic to a map that is induced by an affine map on the Lie group). This is done in two steps -- for \(m=3\) and for \(m \ge 4\). For the computation of these basic invariants of the fixed point theory the authors use some special ``averaging formulas''. It is proved that for \(\text{Nil}_3/\Gamma\) the aforementioned sets are equal correspondingly to \(2 \mathbb{Z}, 2 \mathbb{N} \cup \{ 0 \}\) and \(2 \mathbb{N} \cup \infty\). Also such sets are found for \(\text{ Nil}_3/\Pi\), for \(\text{Nil}_m/\Gamma_m\) and for \(\text{Nil}_m/\Pi (m \ge 4)\).
Reviewer: V. V. Gorbatsevich (Moskva)Existence of two-parameter crossings, with applications.https://www.zbmath.org/1452.570242021-02-12T15:23:00+00:00"Williams, Jonathan D."https://www.zbmath.org/authors/?q=ai:williams.jonathan-dIt is well-known that the image of the critical set of a Morse 2-function from an \(n\)-manifold \(M\) to a \(2\)-manifold \(B\) is an immersed collection of cusped \(1\)-submanifolds. The main goal of this interesting paper is to explain when it is possible to move the critical image around by a homotopy of the fibration map (Theorem 2.4) and to give examples when \(M\) is a closed \(4\)-manifold. Another paper by the author directly connected to this subject is [\textit{J. Williams}, Geom. Topol. 14, No. 2, 1015--1061 (2010; Zbl 1204.57027)].
Reviewer: Dorin Andrica (Riyadh) (MR4117573)Taming the pseudoholomorphic beasts in \(\mathbb{R} \times (S^1 \times S^2)\).https://www.zbmath.org/1452.530742021-02-12T15:23:00+00:00"Gerig, Chris"https://www.zbmath.org/authors/?q=ai:gerig.chrisSummary: For a closed oriented smooth \(4\)-manifold \(X\) with \(b^2_+(X)>0\), the Seiberg-Witten invariants are well-defined. Taubes' ``\( \operatorname{SW}=\operatorname{Gr} \)'' theorem asserts that if \(X\) carries a symplectic form then these invariants are equal to well-defined counts of pseudoholomorphic curves, Taubes' Gromov invariants. In the absence of a symplectic form, there are still nontrivial closed self-dual \(2\)-forms which vanish along a disjoint union of circles and are symplectic elsewhere. This paper and its sequel describe well-defined integral counts of pseudoholomorphic curves in the complement of the zero set of such near-symplectic \(2\)-forms, and it is shown that they recover the Seiberg-Witten invariants over \(\mathbb{Z}/2\mathbb{Z} \). This is an extension of ``\( \operatorname{SW}=\operatorname{Gr} \)'' to nonsymplectic \(4\)-manifolds.
The main result of this paper asserts the following. Given a suitable near-symplectic form \(\omega\) and tubular neighborhood \(\mathcal{N}\) of its zero set, there are well-defined counts of pseudoholomorphic curves in a completion of the symplectic cobordism \((X-\mathcal{N},\omega)\) which are asymptotic to certain Reeb orbits on the ends. They can be packaged together to form ``near-symplectic'' Gromov invariants as a function of spin-c structures on \(X\).Psyquandles, singular knots and pseudoknots.https://www.zbmath.org/1452.570102021-02-12T15:23:00+00:00"Nelson, Sam"https://www.zbmath.org/authors/?q=ai:nelson.sam"Oyamaguchi, Natsumi"https://www.zbmath.org/authors/?q=ai:oyamaguchi.natsumi"Sazdanovic, Radmila"https://www.zbmath.org/authors/?q=ai:sazdanovic.radmilaBiquandles are algebraic structures with axioms motivated by the oriented Reidemeister moves of diagrams of knots and links. Biquandles have been used to define invariants of classical and virtual oriented knots and links and the results of the paper under review are motivated by effectiveness of these structures in distinguishing oriented knots and links. Singular knots and links are 4-valent spatial graphs considered up to rigid vertex isotopy, where a vertex can be thought of as the result of two strands of a knot or link getting glued together in a fixed position. A singular knot or link with exactly one singular crossing is a 2-bouquet graph. Pseudoknots are knots whose diagrams consist of usual crossings and precrossings (classical crossings where one cannot tell which strand goes on top).
The authors generalize the notion of biquandles to psyquandles and use them to define invariants of oriented singular links and pseudolinks. In addition to psyquandle counting invariants, they also introduce Alexander psyquandles and corresponding invariants such as Alexander psyquandle polynomials and Alexander-Gröbner psyquandle invariants of oriented singular knots and links. The authors define a generalization of the Alexander polynomial for oriented singular links and pseudolinks which they refer to as the Jablan polynomial and compute the invariant for all pseudoknots with up to five crossings and all 2-bouquet graphs with up to six classical crossings.
Reviewer: Mahender Singh (Manauli)Operations on stable moduli spaces.https://www.zbmath.org/1452.550122021-02-12T15:23:00+00:00"Galatius, Søren"https://www.zbmath.org/authors/?q=ai:galatius.soren"Randal-Williams, Oscar"https://www.zbmath.org/authors/?q=ai:randal-williams.oscarQuillen was the first to note the phenomenon of homological stability for groups [unpublished, 1971], and the topological version, due to \textit{G. Segal} [Invent. Math. 21, 213--221 (1973; Zbl 0267.55020)], was applied by Harer to moduli spaces of surfaces in [\textit{J. L. Harer}, Ann. Math. (2) 121, 215--249 (1985; Zbl 0579.57005)]. The authors have studied
moduli spaces of higher dimensional manifolds in a series of papers, starting with [\textit{S. Galatius} and \textit{O. Randal-Williams}, Geom. Topol. 14, No. 3, 1243--1302 (2010; Zbl 1205.55007)] and continuing with [\textit{S. Galatius} and \textit{O. Randal-Williams}, J. Am. Math. Soc. 31, No. 1, 215--264 (2018; Zbl 1395.57044); Ann. Math. (2) 186, No. 1, 127--204 (2017; Zbl 1412.57026)]. See also their survey [\textit{O. Randal-Williams} and \textit{S. Galatius}, ``Moduli spaces of manifolds: a user's guide'', in: Handbook of homotopy theory. Boca Raton, FL: CRC Press. 43 p. (2020; Zbl 07303336)].
The question addressed in this paper is to what extent the stabilization is topological -- that is, coming from a map between the moduli space constructed by a topological operation on the manifolds under consideration (as is the case with surfaces, where one takes the connected sum with a torus to increase genus). The authors consider the moduli space \(\mathcal{M}(W)\) of fiber bundles with fiber a given smooth closed \(d\)-manifold \(W\) (that is, \(B\mathrm{Diff}(W)\)), when \(d=2n\), under stable diffeomorphims (defined by taking connected sums with copies of \(S^{n}\times S^{n}\)). More generally, they consider moduli spaces \(\mathcal{M}^{\Lambda}(W,\lambda)\) of manifolds \(W\) with \(\Lambda\)-structures
\(\lambda:\mathrm{Fr}(TW)\to\Lambda\) on their framed tangent bundles (for a suitable space \(\Lambda\)). Their main theorem states that for any abelian group \(A\), there is a canonical isomorphism \(H^{i}(\mathcal{M}^{\Lambda}(W,\lambda);A)\cong H^{i}(\mathcal{M}^{\Lambda}(W',\lambda');A)\) (induced by a zigzag of maps) whenever \(W\) and \(W'\) are simply connected, \(d=2n>4\), and \(i\) is small compared to the (non-zero) Euler characteristics \(\chi(W)\) and \(\chi(W')\) (which must have the same \(p\)-adic valuation for all primes \(p\) not invertible in \(\mathrm{End}_{\mathbb Z}(A)\)). As an example, they deduce that \(H^{i}(\mathcal{M}(W);{\mathbb Z}_{(p)})\cong H^{i}(\mathcal{M}(W');{\mathbb Z}_{(p)})\) for small \(i\), if \(\chi(W)\) and \(\chi(W')\) have the same \(p\)-adic valuation.
In the works cited above the authors had previously produced maps \(\mathcal{M}^{\Lambda}(W,\lambda)\to(\Omega^{\infty}MT\Theta)/\!/\mathrm{hAut}(u)\) which induce isomorphisms in homology in a range, where \(u:\Theta\to\Lambda\) is a cofibrant \(\mathrm{GL}_{d}({\mathbb R})\)-equivariant fibration and \(MT\Theta\) is the Thom spectrum of the inverse of the corresponding \({\mathbb R}^{d}\)-bundle on the Borel construction
\(B=\Theta/\!/\mathrm{GL}_{d}({\mathbb R})\). The main theorem here is then proved by constructing certain operations
\(\psi^{q}:\Omega_{k}^{\infty}MT\Theta\to\Omega_{qk}^{\infty}MT\Theta\), coming from self-maps of the Thom spectra, for any \(q,k\in{\mathbb Z}\) (with \(2\) inverted when \(q=2\)). Here \(\Omega_{k}^{\infty}MT\Theta\) is a union of certain path components.
Reviewer: David Blanc (Haifa)On the center of the group of quasi-isometries of the real line.https://www.zbmath.org/1452.200362021-02-12T15:23:00+00:00"Chakraborty, Prateep"https://www.zbmath.org/authors/?q=ai:chakraborty.prateepSummary: Let \(QI ( \mathbb{R} )\) denote the group of all quasi-isometries \(f : \mathbb{R} \rightarrow \mathbb{R} \). Let \(Q_+\)(and \(Q_-)\) denote the subgroup of \(QI ( \mathbb{R} )\) consisting of elements which are identity near \(- \infty \) (resp. \(+ \infty )\). We denote by \(QI^+(\mathbb{R} )\) the index 2 subgroup of \(QI ( \mathbb{R} )\) that fixes the ends \(+ \infty, - \infty \). We show that \(QI^+(\mathbb{R}) \cong Q_+ \times Q_-\). Using this we show that the center of the group \(QI ( \mathbb{R} )\) is trivial.Admissible topologies for groups of homeomorphisms and substitutions of groups of \(G\)-spaces.https://www.zbmath.org/1452.540222021-02-12T15:23:00+00:00"Karassev, A."https://www.zbmath.org/authors/?q=ai:karassev.alexandre"Kozlov, K. L."https://www.zbmath.org/authors/?q=ai:kozlov.konstantin-lThe group of homeomorphisms in the discrete topology acts continuously and transitively on a homogeneous space \(X\). This paper aims at substituting the discrete acting group of a \(G\)-space by the quotient group with respect to the kernel of action, which would preserve various properties (transitivity, being a coset space, preserving a fixed equuniformity in case of \(G\)-Tychonoff space) of actions (Theorem 3.1). Such substitution is interesting from the perspective of investigating metrizability of phase spaces (Theorem 3.27) and deriving bounds on cardinal invariants of acting groups with admissible topologies (Theorem 3.12, 3.13). The authors exploit the product structure of the class of \(\mathcal{G}\)-range topological groups and the presence of a uniform structure on \(G\)-spaces to achieve their goals.
Reviewer: Kateryna Pavlyk (Tartu)Topological folding on the chaotic projective spaces and their fundamental group.https://www.zbmath.org/1452.510032021-02-12T15:23:00+00:00"Abu-Saleem, M."https://www.zbmath.org/authors/?q=ai:abu-saleem.mohammed"Al-Omeri, W. Faris."https://www.zbmath.org/authors/?q=ai:al-omeri.wadei-farisSummary: In this article we will introduce different types of topological foldings on chaotic projective space. The limit of topological foldings on the fundamental group of the real projective plane will be obtained. The chain of folding on the chaotic projective spaces will induce a chain of fundamental groups. The relations between these chains will be achieved.Möbius invariant metrics on the space of knots.https://www.zbmath.org/1452.570082021-02-12T15:23:00+00:00"O'Hara, Jun"https://www.zbmath.org/authors/?q=ai:ohara.junSummary: We give a condition for a function to produce a Möbius invariant weighted inner product on the tangent space of the space of knots, and show that some kind of Möbius invariant knot energies can produce Möbius invariant and parametrization invariant weighted inner products. They would give a natural way to study the evolution of knots in the framework of Möbius geometry.Cohomogeneity one manifolds with singly generated rational cohomology.https://www.zbmath.org/1452.570222021-02-12T15:23:00+00:00"DeVito, Jason"https://www.zbmath.org/authors/?q=ai:devito.jason"Kennard, Lee"https://www.zbmath.org/authors/?q=ai:kennard.leeLet \({\mathbb{Q}}{\mathbb{P}}^n_k\) denote
any smooth, simply connected, closed manifold whose rational cohomology is isomorphic to
\({\mathbb{Q}}[x]/(x^{n+1})\) where the generator \(x\) has degree \(k\). If \(k\) is odd, then such a manifold is
a rational sphere or a point. Other typical examples of such manifolds are simply connected, closed manifolds
with the rational cohomology of a compact rank one symmetric space. Indeed, the authors call
the parameters \((n,k)\) standard, if they correspond to a rank one symmetric space.
It would be interesting to find highly symmetric models for \({\mathbb{Q}}{\mathbb{P}}^n_k\)
with non-standard parameters. The authors provide a negative result: They show that if a
\({\mathbb{Q}}{\mathbb{P}}^n_k\) admits a cohomogeneity one action, then \(n\) and \(k\) are standard.
Moreover, if \(k\) is even, then the space is diffeomorphic to a rank one symmetric space, the Grassmannian
\(\mathrm{SO}(2m+1)/\mathrm{SO}(2)\times \mathrm{SO}(2m-1)\) or \(\mathrm{G}_2/\mathrm{SO}(4)\).
Biquotients with four-periodic rational cohomology were classified by the first author in
[Geom. Dedicata 195, 121--135 (2018; Zbl 1402.57025)].
The second main result of this paper is a step toward an analogous classification for cohomogeneity
one manifolds: A simply connected, closed manifold with the rational cohomology of
\({\mathbb{S}}^2\times {\mathbb{H}}{\mathbb{P}}^n\) admits a cohomogeneity one action if
and only if it is diffeomorphic to \({\mathbb{S}}^2\times {\mathbb{H}}{\mathbb{P}}^n\),
\({\mathbb{S}}^2\times (\mathrm{G}_2/\mathrm{SO}(4))\), or the unique linear non-trivial
\({\mathbb{H}}{\mathbb{P}}^n\) bundle over \({\mathbb{S}}^2\).
Reviewer: Marja Kankaanrinta (Helsinki)Realizable ranks of joins and intersections of subgroups in free groups.https://www.zbmath.org/1452.200192021-02-12T15:23:00+00:00"Soroko, Ignat"https://www.zbmath.org/authors/?q=ai:soroko.ignatAuthor's abstract: The famous Hanna Neumann Conjecture (now the Friedman-Mineyev theorem) gives an upper bound for the ranks of the intersection of arbitrary subgroups \(H\) and \(K\) of a non-abelian free group. It is an interesting question to ``quantify'' this bound with respect to the rank of \(H \vee K\), the subgroup generated by \(H\) and \(K\). We describe a set of realizable values \((rk(H \vee K), rk(H \cap K))\) for arbitrary \(H\), \(K\), and conjecture that this locus is complete. We study the combinatorial structure of the topological pushout of the core graphs for \(H\) and \(K\) with the help of graphs introduced by Dicks in the context of his Amalgamated Graph Conjecture. This allows us to show that certain conditions on ranks of \( H \vee K\), \(H \cap K\) are not realizable, thus resolving the remaining open case \(m = 4\) of Guzman's ``Group-Theoretic Conjecture'' in the affirmative. This in turn implies the validity of the corresponding ``Geometric Conjecture'' on hyperbolic \(3\)-manifolds with a \(6\)-free fundamental group. Finally, we prove the main conjecture describing the locus of realizable values for the case when \(rk(H) = 2\).
Reviewer: Alexander Ivanovich Budkin (Barnaul)Toward an algebraic Donaldson-Floer theory.https://www.zbmath.org/1452.140572021-02-12T15:23:00+00:00"Li, Jun"https://www.zbmath.org/authors/?q=ai:li.jun.14|li.jun.2|li.jun.10|li.jun.11|li.jun.12|li.jun|li.jun.3|li.jun.13|li.jun.6|li.jun.7|li.jun.8|li.jun.1Summary: We construct the relative Donaldson polynomial invariants of a pair of a smooth divisor in a smooth surface, taking values in an operational algebraic Floer homology group. We conjecture that this pair forms an algebraic Donaldson-Floer theory.
For the entire collection see [Zbl 1400.53003].Dehn surgery on knots in \(S^3\) producing Nil Seifert fibered spaces.https://www.zbmath.org/1452.570142021-02-12T15:23:00+00:00"Ni, Yi"https://www.zbmath.org/authors/?q=ai:ni.yi"Zhang, Xingru"https://www.zbmath.org/authors/?q=ai:zhang.xingruLet \(S_K^3(p/q)\) denote the 3-manifold obtained by Dehn surgery with slope \(p/q\) along a knot \(K\) in the 3-sphere. The main result of the paper is the following. Suppose that \(K\) is not the trefoil knot; if \(S_K^3(p/q)\) is a Seifert fibered space admitting the Nil geometry (i.e., with Euler number \(\ne 0\) and with Euclidean base orbifold) then \(q=1\) and \(p\) is one of the numbers 60, 144, 156, 288, 300, and \(K\) is either a hyperbolic knot or a cable over the trefoil knot. Previously, \textit{S. Boyer} proved in [Topology Appl. 121, No. 3, 383--413 (2002; Zbl 1021.57011)] that there is at most one surgery on a hyperbolic knot which can produce a Seifert fibered space with the Nil geometry, and that in such a case the surgery slope is integer (\(q = 1\)).
``The main ingredient of the method is the use of the correction terms (also known as the \(d\)-invariants) for rational homology spheres together with their \(\mathrm{Spin}^c\) structures, defined in a paper by \textit{P. Ozsváth} and \textit{Z. Szabó} [Adv. Math. 173, No. 2, 179--261 (2003; Zbl 1025.57016)]''.
For the entire collection see [Zbl 1437.55002].
Reviewer: Bruno Zimmermann (Trieste)Tangential vector fields on Kendall shape space.https://www.zbmath.org/1452.530282021-02-12T15:23:00+00:00"Mtibaa, Riadh"https://www.zbmath.org/authors/?q=ai:mtibaa.riadh"Khan, Salam"https://www.zbmath.org/authors/?q=ai:khan.salamSummary: We establish special expressions of the horizontal lifts of some tangential vector fields on the Kendall shape space denoted by \(\Sigma^k_m\). The latter helps to associate shapes to subsets of real number matrices with dimensions \(m \times k\), where \(m\geq 3\) and \(k\geq m + 2\). The shape results from the application of a specific procedure that eliminates the effects of translation, volume, and rotation from the initial matrices. So, the Kendall shape space appears as the quotient of the unit sphere \(\mathcal{S}^k_m\) in \(\mathbb{R}^{m(k-1)-1}\) modulo the special orthogonal group \(\mathrm{SO}(m)\). We use particular curves in \(\mathcal{S}^k_m\), as well as the chain rule in order to calculate special tangential vector fields on \(\mathcal{S}^k_m\) that we exploit to deduce the horizontal lifts of the sought tangential vector fields on the Kendall shape space \(\Sigma^k_m\).Resolutions for twisted tensor products.https://www.zbmath.org/1452.160032021-02-12T15:23:00+00:00"Shepler, Anne"https://www.zbmath.org/authors/?q=ai:shepler.anne-v"Witherspoon, Sarah"https://www.zbmath.org/authors/?q=ai:witherspoon.sarah-jThe twisted tensor product \(A\otimes_\tau B\), as in [\textit{A. Čap} et al., Commun. Algebra 23, No. 12, 4701--4735 (1995; Zbl 0842.16005)], is the tensor product \(A\otimes_k B\) (of associative algebras over a commutative ring \(k\)) equipped with a ring multiplication determined by a bijective linear map \(\tau\colon B\otimes_k A\rightarrow A\otimes_k B\), satisfying certain conditions, known as twisting map. In the paper under review, \(k\) is assumed to be a field. Ore extensions, skew group algebras, Weyl algebras are among the many examples of twisted tensor products. The main goal of the paper under review is two-fold:
\begin{itemize}
\item[1.] Given an \(A\)-bimodule \(M\) and a \(B\)-bimodule \(N\), the authors construct a projective resolution of \(M\otimes_kN\) as bimodule over \(A\otimes_\tau B\) from an \(A\)-bimodule and a \(B\)-bimodule projective resolutions of \(M\) and \(N\), respectively, under certain compatibility conditions involving \(\tau\).
\item[2.] Given a (left) \(A\)-module \(M\) and a (left) \(B\)-module \(N\), the authors construct a \(A\otimes_\tau B\)-projective resolution of \(M\otimes_k N\) knowing left \(A\) and \(B\)-projective resolutions of \(M\) and \(N\), respectively, under certain compatibility conditions involving \(\tau\).
\end{itemize}
In particular, the authors show that twisted product resolutions of \(A\otimes_\tau B\) as bimodule always exist. In many examples, some of the compatibility conditions can be dropped. Several resolutions in different setups arise as particular cases of these results given by the authors. Using these general constructions, the authors construct twist product resolutions for bimodules and modules over the particular case of Ore extensions by regarding them as certain twisted tensor products giving special emphasis to iterated Ore extensions.
Reviewer: Tiago Cruz (Stuttgart)Isotopies of surfaces in 4-manifolds via banded unlink diagrams.https://www.zbmath.org/1452.570172021-02-12T15:23:00+00:00"Hughes, Mark C."https://www.zbmath.org/authors/?q=ai:hughes.mark-c"Kim, Seungwon"https://www.zbmath.org/authors/?q=ai:kim.seungwon"Miller, Maggie"https://www.zbmath.org/authors/?q=ai:miller.maggieIn this paper, the authors study a generalization of banded unlink diagrams to embedded surfaces in an arbitrary 4-manifold equipped with a Morse function, extending the work of \textit{F. J. Swenton} [J. Knot Theory Ramifications 10, No. 8, 1133--1141 (2001; Zbl 1001.57044)] and \textit{C. Kearton} and \textit{V. Kurlin} [Algebr. Geom. Topol. 8, No. 3, 1223--1247 (2008; Zbl 1151.57027)] for the case of \(S^4\). A band for a link \(L\) in a 3-manifold is an embedded 2-disk attached to \(L\) along two segments, and a banded unlink is the union of an unlink \(L\) and a family of disjoint bands for \(L\). The authors describe an embedded surface in a 4-manifold \(X\) by arranging it in banded unlink position, which satisfies that the intersections with the level sets of a given Morse function consist of disjoint unions of embedded disks and banded unlinks. And they represent a surface in banded unlink position by a banded unlink diagram, which is defined in terms of the Kirby diagram in \(S^3\) associated with \(X\), equipped with an unlink \(L\) and a family of bands for \(L\).
The main results are as follows. The authors describe a set of moves on banded unlink diagrams, called band moves. Band moves consist of moves established by \textit{K. Yoshikawa} [Osaka J. Math. 31, No. 3, 497--522 (1994; Zbl 0861.57033)] for the case of \(S^4\), and several additional moves. They show that for a smooth 4-manifold \(X\) with a Kirby diagram, the embedded surfaces \(\Sigma\) and \(\Sigma'\) in \(X\) are isotopic if and only if their banded unlink diagrams are related by a finite sequence of band moves. This is shown by case analysis and using for some cases an argument analogous to the one due to Kearton and Kurlin [loc. cit.]. Using this result, the authors consider an embedded surface \(\Sigma\) via a bridge trisection of \(\Sigma\), and they affirmatively prove a conjecture of \textit{J. Meier} and \textit{A. Zupan} [Proc. Natl. Acad. Sci. USA 115, No. 43, 10880--10886 (2018; Zbl 1418.57017)]: for surfaces \(S\) and \(S'\) in bridge position with respect to a trisection \(\mathcal{T}\) of a closed 4-manifold \(X\), if \(S\) and \(S'\) are isotopic, then they are related by a sequence of perturbations and deperturbations, followed by a \(\mathcal{T}\)-regular isotopy. Further, they consider the case of unit surfaces in \(\mathbb{C}P^2\). A unit surface is a surface in \(\mathbb{C}P^2\) which intersects the standard \(\mathbb{C}P^1 \subset \mathbb{C}P^2\) in exactly one pont. In particular, they show that for a genus-\(g\) unit surface-knot \(F=S \# \mathbb{C}P^1 \subset \mathbb{C}P^2\), where \(S \subset S^4\) is an orientable surface that is \(0\)-concordant to a band-sum of twist-spun knots and unknotted surfaces, \(F\) is isotopic to the standard \(\mathbb{C}P^1\) trivially stabilized \(g\) times.
Reviewer: Inasa Nakamura (Kanazawa)On balanced planar graphs, following W. Thurston.https://www.zbmath.org/1452.570202021-02-12T15:23:00+00:00"Koch, Sarah"https://www.zbmath.org/authors/?q=ai:koch.sarah-c"Lei, Tan"https://www.zbmath.org/authors/?q=ai:lei.tanA branched covering \(f:S^2 \to S^2\) of degree \(d \ge 2\) is \textit{generic} if \(f\) has exactly \(2d - 2\) ramification or critical values; the preimage of a Jordan curve running through these \(2d-2\) critical values can be considered as a 4-valent graph in \(S^2\), and the basic question addressed in the present paper is: which 4-valent graphs in \(S^2\) can be realized in such a way? (in the context of this question, Thurston asked: What is the shape of a rational map?) The main result of the present paper is a complete characterization of these \textit{balanced} graphs, following a proof suggested by Thurston (after presenting a complete list of such graphs for the case of degree \(d = 4\)).
For the entire collection see [Zbl 1437.55002].
Reviewer: Bruno Zimmermann (Trieste)On geometry of foliations of codimension 1.https://www.zbmath.org/1452.530242021-02-12T15:23:00+00:00"Bayturaev, A. M."https://www.zbmath.org/authors/?q=ai:bayturaev.a-mAuthor's abstract: In this paper, we examine geometry and topology of foliations generated by level surfaces of metric functions.
Reviewer: Georges Habib (Beirut)On geometry of vector fields.https://www.zbmath.org/1452.370202021-02-12T15:23:00+00:00"Narmanov, A. Ya."https://www.zbmath.org/authors/?q=ai:narmanov.abdigappar-yakubovich|narmanov.abdugappar-yakubovich"Saitova, S. S."https://www.zbmath.org/authors/?q=ai:saitova.s-sThe authors present results on the geometry of the attainability set of a family of vector fields, results regarding the geometry of \(T\)-attainability sets and the geometry of orbits of Killing vector fields. This paper contains some interesting findings which could provide inspiration for further studies on the subject.
Reviewer: Themistocles M. Rassias (Athína)An important step for the computation of the HOMFLYPT skein module of the Lens spaces \(L(p, 1)\) via braids.https://www.zbmath.org/1452.570052021-02-12T15:23:00+00:00"Diamantis, Ioannis"https://www.zbmath.org/authors/?q=ai:diamantis.ioannis"Lambropoulou, Sofia"https://www.zbmath.org/authors/?q=ai:lambropoulou.sofiaIn this paper the authors build on their previously established braid theoretic approach of deriving the Homflypt skein module of the lens spaces \(L(p,1)\) from its counterpart for the solid torus. Starting from two different bases for the skein module of the solid torus \(\mathcal{S}(ST)\), two different spanning sets can be obtained \(\Lambda ^{\prime\mathrm{aug}}\) and \(\Lambda^{\mathrm{aug}}\). Here they focus on the latter spanning set and they show that in order to obtain the Homflypt skein module for \(L(p,1)\), it suffices to do the following. First, consider braid band moves only on the first moving strand of the elements in \(\Lambda ^{\mathrm{aug}}\). Then, impose the invariant \(X\), which is the analogue of the Homflypt polynomial in the case of Coxeter groups of type B, and solve the derived infinite system of equations.
Reviewer: Dimos Goundaroulis (Lausanne)A decade of Thurston stories.https://www.zbmath.org/1452.570022021-02-12T15:23:00+00:00"Sullivan, Dennis"https://www.zbmath.org/authors/?q=ai:sullivan.dennis-m|sullivan.dennis-pThis is a collection of historical and mathematical anecdotes around Thurston from the years 1971--1983 (first published in the Notices of the AMS). In a recent postscript from 2019, the author added:
``For me personally, ``What's next'' means the following question, given that 3-manifolds have the very specific structure predicted and developed by Bill Thurston almost forty years ago: \textit{Why are 3-manifolds the way they are?} Namely, one wonders what unexpected and beautiful significance and/or applications does this very specific structure mean for other parts of mathematics, physics, or science in general.''
For the entire collection see [Zbl 1437.55002].
Reviewer: Bruno Zimmermann (Trieste)An invitation to coherent groups.https://www.zbmath.org/1452.570192021-02-12T15:23:00+00:00"Wise, Daniel T."https://www.zbmath.org/authors/?q=ai:wise.daniel-tA group is \textit{coherent} if every finitely generated subgroup has a finite presentation. The motivation comes from low-dimensional topology, in particular free groups, surface groups and 3-manifold groups are coherent. The present paper is an extensive survey on coherent groups, finishing with a list of 41 problems and 10 pages of references, and confronting coherence with various other concepts in combinatorial and geometric group theory. ``A primary motivation for me stems
from the desire to develop a class of groups that are close in nature to the fundamental groups of 3-manifolds, and echo one of their most salient properties.''
``There is a tremendous desire in mathematics to understand and classify all possible forms of some family of objects, and our human condition makes 3-manifolds appear rather quickly on the list of priorities. Nonetheless, I don't think 3-manifolds are ``for'' something else in mathematics, and I am unconvinced about how frequently they appear as crucial objects outside their domain -- in contrast to surfaces which are comparitively ubiquitous in mathematics. Instead, 3-manifolds are a topological ultimate goal, and I likewise hope coherent groups are an algebraic telos, that might motivate significant and useful technic.''
For the entire collection see [Zbl 1437.55002].
Reviewer: Bruno Zimmermann (Trieste)Equivariant \(K\)-theory of divisive torus orbifolds.https://www.zbmath.org/1452.550062021-02-12T15:23:00+00:00"Sarkar, Soumen"https://www.zbmath.org/authors/?q=ai:sarkar.soumenTorus orbifolds are generalizations of toric orbifolds. In the paper under review it is defined what it means for a torus orbifold to be divisive. For divisive torus orbifolds torus invariant cell structures are constructed. This construction is used to compute certain generalized cohomologies of divisive torus orbifolds. This includes \(K\)-theory, unitary bordism and ordinary cohomology.
The discussion is motivated by the case of divisive weighted projective spaces which was considered in [\textit{M. Harada} et al., Tohoku Math. J. (2) 68, No. 4, 487--513 (2016; Zbl 1360.55006)].
Reviewer: Michael Wiemeler (Münster)The rational homology ring of the based loop space of the gauge groups and the spaces of connections on a four-manifold.https://www.zbmath.org/1452.550142021-02-12T15:23:00+00:00"Terzić, S."https://www.zbmath.org/authors/?q=ai:terzic.svjetlanaThis paper mainly contains two parts. In Section 2 the author discusses the ranks of the homotopy groups of a simply connected closed 4-dimensional manifold \(M\). She reviews known results on \(\pi_*(M)\) and proves that the ranks of \(\pi_*(M)\) depend only on the second Betti number \(b_2(M)\) of \(M\) by constructing a minimal free Lie model of \(M\).
In Section 3 the author considers the rational homology ring of gauge groups and related spaces. Let \(G\) be a compact, semisimple, simply connected Lie group and let \(\pi:P\to M\) be a principal \(G\)-bundle over \(M\). Then the gauge group \(\mathcal{G}\) is the topological group of \(G\)-equivariant automorphisms of \(P\) that induce the identity map on \(M\), and the pointed gauge group \(\mathcal{G}_0\) is the subgroup of automorphisms that restrict to the identity map on the fiber over the base point of \(M\). According to Atiyah-Bott and Gottlieb, \(\mathcal{G}_0\simeq\text{Map}_*(M,G)\). Denote the space of connections on \(P\) by \(\mathcal{A}\), the space of irreducible connections by \(\mathcal{A}^*\) and the center of \(G\) by \(Z(G)\). Let \(\tilde{\mathcal{B}}=\mathcal{A}/\mathcal{G}_0\) and \(\mathcal{B}^*=\mathcal{A}^*/(\mathcal{G}/Z(G))\). Assuming \(\mathcal{G}\) is connected, the author applies the Milnor-Moore Theorem to obtain the rational homology rings \(H_*(\Omega\tilde{\mathcal{B}};\mathbb{Q})\) and \(H_*(\Omega\mathcal{B}^*;\mathbb{Q})\), which depend only on \(b_2(M)\) and \(\pi_*(G)\).
Finally, as examples the author calculates \(H_*(\Omega\tilde{\mathcal{B}};\mathbb{Q})\) and \(H_*(\Omega\mathcal{B}^*;\mathbb{Q})\) for the cases when \(P\) is a principal \(SU(2)\)-bundle over a 4-manifold and a principal \(SU(3)\)-bundle over a spin 4-manifold.
Reviewer: Tseleung So (Regina)Construction of subsurfaces via good pants.https://www.zbmath.org/1452.570012021-02-12T15:23:00+00:00"Liu, Yi"https://www.zbmath.org/authors/?q=ai:liu.yi"Markovic, Vladimir"https://www.zbmath.org/authors/?q=ai:markovic.vladimirThe good pants technology is a systematic method to produce surface subgroups in cocompact lattices of \(\mathrm{PSL}_2(\mathbb C)\) and \(\mathrm{PSL}_2(\mathbb R)\) (using frame flow). The \(\mathrm{PSL}_2(\mathbb C)\)-case leads to a proof of the Surface Subgroup Conjecture [\textit{J. Kahn} and \textit{V. Markovic}, Ann. Math. (2) 175, No. 3, 1127--1190 (2012; Zbl 1254.57014)], and the \(\mathrm{PSL}_2(\mathbb R)\)-case to a proof of the Ehrenpreis Conjecture on Riemann surfaces [\textit{J. Kahn} and \textit{V. Markovic}, in: Proceedings of the International Congress of Mathematicians (ICM 2014), Seoul, Korea, August 13--21, 2014. Vol. II: Invited lectures. Seoul: KM Kyung Moon Sa. 897--909 (2014; Zbl 1373.57037)].
The \textit{Surface Subgroup Conjecture} asserts that any cocompact lattice in \(\mathrm{PSL}_2(\mathbb C)\) contains a surface subgroup, in particular any closed hyperbolic 3-manifold contains \(\pi_1\)-injective immersed subsurfaces (this plays a fundamental role in the resolution of the \textit{Virtual Haken Conjecture} for hyperbolic 3-manifolds).
The \textit{Ehrenpreis Conjecture} asserts that, for any two closed Riemann surfaces of the same genus and positive constant \(\epsilon > 0\), there exists a \((1+ \epsilon)\)-quasiconformal map between some finite covers of the two surfaces; equivalently, any two closed Riemann surfaces are virtually \((1+ \epsilon)\)-bilipschitz equivalent, for any positive constant \(\epsilon\).
The present paper is a survey on the constructions and methods leading to the solutions of these two conjectures, and also on some further applications of the methods.
For the entire collection see [Zbl 1437.55002].
Reviewer: Bruno Zimmermann (Trieste)Some comments on Laakso graphs and sets of differences.https://www.zbmath.org/1452.540192021-02-12T15:23:00+00:00"Margaris, Alexandros"https://www.zbmath.org/authors/?q=ai:margaris.alexandros"Robinson, James C."https://www.zbmath.org/authors/?q=ai:robinson.james-c|robinson.james-c.1Summary: We recall a variation of a construction due to \textit{T. J. Laakso} [Bull. Lond. Math. Soc. 34, No. 6, 667--676 (2002; Zbl 1029.30014)], also used by \textit{U. Lang} and \textit{C. Plaut} [Geom. Dedicata 87, No. 1--3, 285--307 (2001; Zbl 1024.54013)] of a doubling metric space \(X\) that cannot be embedded into any Hilbert space. We give a more concrete version of this construction and motivated by the results of \textit{E. J. Olson} and \textit{J. C. Robinson} [Trans. Am. Math. Soc. 362, No. 1, 145--168 (2010; Zbl 1197.54045)], we consider the Kuratowski embedding \(\Phi(X)\) of \(X\) into \(L^{\infty}(X)\) and prove that \(\Phi (X)-\Phi (X)\) is not doubling.Groups of homeomorphisms of one-manifolds. I: Actions of nonlinear groups.https://www.zbmath.org/1452.370332021-02-12T15:23:00+00:00"Farb, Benson"https://www.zbmath.org/authors/?q=ai:farb.benson"Franks, John"https://www.zbmath.org/authors/?q=ai:franks.john-mThe authors consider finitely presented infinite groups acting by homeomorphisms or diffeomorphisms on \(S^1\), the interval \([0,1]\) or the reals. It is known that any \(C^1\)-action on \(S^1\) of a lattice in a simple Lie group of \(\mathbb R\)-rank at least two factors through a finite group. In the present paper, the authors discuss three basic examples of nonlinear groups: mapping class groups of surfaces, (outer) automorphism groups of free groups, and Baumslag-Solitar groups. Since the paper was written in 2001, at the end of the introduction they give some references to more recent work on such types of problems. For the entire collection see [Zbl 1437.55002].
For Part II and III of this paper see [\textit{B. Farb} and \textit{J. Franks}, Trans. Am. Math. Soc. 355, No. 11, 4385--4396 (2003; Zbl 1025.37024); Ergodic Theory Dyn. Syst. 23, No. 5, 1467--1484 (2003; Zbl 1037.37020)].
Reviewer: Bruno Zimmermann (Trieste)