Recent zbMATH articles in MSC 57Rhttps://www.zbmath.org/atom/cc/57R2022-05-16T20:40:13.078697ZWerkzeugOn Gauss-Bonnet and Poincaré-Hopf type theorems for complex \(\partial\)-manifoldshttps://www.zbmath.org/1483.140082022-05-16T20:40:13.078697Z"Corrêa, Maurício"https://www.zbmath.org/authors/?q=ai:correa-barros.mauricio-jun"Lourenço, Fernando"https://www.zbmath.org/authors/?q=ai:lourenco.fernando"Machado, Diogo"https://www.zbmath.org/authors/?q=ai:machado.diogo"Ferreira, Antonio M."https://www.zbmath.org/authors/?q=ai:ferreira.antonio-mLet \(X\) be a compact complex manifold and let \(D\subset X\) be a reduced reducible divisor with isolated singularities. The authors analyze the case where \(D\) is a union of two (not necessarily irreducible) components \(D_1\) and \(D_2\) such that \(C = D_1\cap D_2\) is a codimension 2 subspace of \(X\) and \(D_1\), \(D_2\) and \(C\) have at most isolated singularities. Under these assumptions, analogues of the classical Gauss-Bonnet and Poincaré-Hopf formulas for the \(\partial\)-manifold \(\widetilde X=X-D\) are obtained. In particular, it turns out that \(\int_X c_n(\Omega_X(\log D))\) is equal to the alternating sum of the Euler characteristic \(\chi(\widetilde X)\) and the Milnor numbers of singular points of \(D_1\), \(D_2\) and \(C\). Similarly, for a given holomorphic vector field \(\mathcal V\) on \(X\) with isolated singularities and logarithmic along \(D\), they show that the Euler characteristic \(\chi(\widetilde X)\) can be represented as a linear combination with coefficients \(\pm 1\) of the Poincaré-Hopf and GSV indices at the singular points of \(\mathcal V\) on \(X\), and the Milnor numbers of the singularities \(D_1\), \(D_2\) and \(C\).
The authors emphasize that they actually modify or generalize a number of previously obtained results for divisors with normal crossings [\textit{S. Iitaka}, J. Fac. Sci., Univ. Tokyo, Sect. I A 23, 525--544 (1976; Zbl 0342.14017); \textit{Y. Norimatsu}, Proc. Japan Acad., Ser. A 54, 107--108 (1978; Zbl 0433.32013); \textit{R. Silvotti}, Invent. Math. 126, No. 2, 235--248 (1996; Zbl 0882.32022); \textit{P. Aluffi}, Trans. Am. Math. Soc. 351, No. 10, 3989--4026 (1999; Zbl 0972.57015)].
Reviewer: Aleksandr G. Aleksandrov (Moskva)Surjectivity of certain adjoint operators and applicationshttps://www.zbmath.org/1483.170162022-05-16T20:40:13.078697Z"Cherifi Hadjiat, Amina"https://www.zbmath.org/authors/?q=ai:cherifi-hadjiat.amina"Lansari, Azzeddine"https://www.zbmath.org/authors/?q=ai:lansari.azzeddineLet \(E\) be the Lie-Fréchet space that consists of all vector fields \(X\) of class \(C^{\infty}\) on \(\mathbb R^{n}\) with a graduation of seminorms \(\| X\| _{r}=\sup _{x\in \mathbb R^n}\max _{k+|\alpha | \leq r}\| D^{\alpha} X(x)\|(1+\| x\|^{2})^{k/2}\) (such Lie algebras were considered by [\textit{R. S. Hamilton}, Bull. Am. Math. Soc., New Ser. 7, 65--222 (1982; Zbl 0499.58003)]. The authors study a subalgebra \(U\) of the Lie algebra \(E\) consisting of all vector fields of the form \(Y_0=X_{0}^{+}+X_{0}^{-}+Z_0\) such that \(X_0(x, y)=A(x, y)=(A^{-}(x), A^{+}(y)),\) with \(A^{-}\) (respectively, \(A^{+})\) a symmetric matrix with eigenvalues \(\lambda<0\) (respectively, \(\lambda >0\)) and \(Z_0\) are germs infinitely flat at the origin. The main result of the paper: the subalgebra \(U\) of the Lie algebra \(E\) admits a hyperbolic structure for the diffeomorphism \(\psi _{t^{\star}}=(\exp{tY_0})_{\star}.\) As an application of this result it was proved that for any admissible subalgebra \(U\) of finite codimension of the Lie algebra \(E\) that has a hyperbolic structure for the flow and satisfies some conditions it holds \(U=E.\)
Reviewer: Anatoliy Petravchuk (Kyiv)Neighborhoods of rational curves without functionshttps://www.zbmath.org/1483.320092022-05-16T20:40:13.078697Z"Falla Luza, Maycol"https://www.zbmath.org/authors/?q=ai:falla-luza.maycol"Loray, Frank"https://www.zbmath.org/authors/?q=ai:loray.frankSummary: We prove the existence of (non compact) complex surfaces with a smooth rational curve embedded such that there does not exist any formal singular foliation along the curve. In particular, at arbitrary small neighborhood of the curve, any meromorphic function is constant. This implies that the Picard group is not countably generated.Nilpotent Cantor actionshttps://www.zbmath.org/1483.370162022-05-16T20:40:13.078697Z"Hurder, Steven"https://www.zbmath.org/authors/?q=ai:hurder.steven-e"Lukina, Olga"https://www.zbmath.org/authors/?q=ai:lukina.olgaA nilpotent Cantor action is a minimal equicontinuous action of a finitely generated group \(\Gamma\) on a Cantor space \(X\), where \(\Gamma\) contains a finitely generated nilpotent subgroup \(\Gamma_0\) of finite index. Nilpotent Cantor actions arise in the classification of renormalizable groups; that is, finitely generated groups which admit a proper self-embedding with image of finite index. These groups arise in the study of laminations with the shape of a compact manifold, and in the classification of generalized Hirsch foliations.
The authors prove that any effective or faithful action of a finitely generated group on a Cantor space, which is continuously orbit equivalent to a nilpotent Cantor action, must itself be a nilpotent Cantor action. As an application, they obtain new invariants of nilpotent Cantor actions under continuous orbit equivalence. The virtual nilpotency class \(vc(\Gamma)\) of a finitely generated virtually nilpotent group \(\Gamma\) is defined as the length of a central series for a torsion-free nilpotent subgroup of finite index. The second main result is that two finitely generated groups admitting continuously orbit equivalent effective Cantor actions have the same virtual nilpotency class.
Reviewer: Mahender Singh (Sahibzada Ajit Singh Nagar)Geometry of lightlike locus on mixed type surfaces in Lorentz-Minkowski 3-space from a contact viewpointhttps://www.zbmath.org/1483.530342022-05-16T20:40:13.078697Z"Honda, Atsufumi"https://www.zbmath.org/authors/?q=ai:honda.atsufumi"Izumiya, Shyuichi"https://www.zbmath.org/authors/?q=ai:izumiya.shyuichi"Saji, Kentaro"https://www.zbmath.org/authors/?q=ai:saji.kentaro"Teramoto, Keisuke"https://www.zbmath.org/authors/?q=ai:teramoto.keisukeSummary: A surface in the Lorentz-Minkowski 3-space is generally a mixed type surface, namely, it has the lightlike locus. We study local differential geometric properties of such a locus on a mixed type surface. We define a frame field along a lightlike locus, and using it, we define two lightlike ruled surfaces along a lightlike locus which can be regarded as lightlike approximations of the surface along the lightlike locus. We study a relationship of singularities of these lightlike surfaces and differential geometric properties of the lightlike locus. We also consider the intersection curve of two lightlike approximations, which gives a model curve of the lightlike locus.Twistors, self-duality, and \(\text{spin}^c\) structureshttps://www.zbmath.org/1483.530712022-05-16T20:40:13.078697Z"LeBrun, Claude"https://www.zbmath.org/authors/?q=ai:lebrun.claude-rSummary: The fact that every compact oriented 4-manifold admits \(\text{spin}^c\) structures was proved long ago by Hirzebruch and Hopf. However, the usual proof is neither direct nor transparent. This article gives a new proof using twistor spaces that is simpler and more geometric. After using these ideas to clarify various aspects of four-dimensional geometry, we then explain how related ideas can be used to understand both spin and \(\text{spin}^c\) structures in any dimension.Hamiltonian circle actions with almost minimal isolated fixed pointshttps://www.zbmath.org/1483.530922022-05-16T20:40:13.078697Z"Li, Hui"https://www.zbmath.org/authors/?q=ai:li.hui.3|li.hui|li.hui.2|li.hui.4|li.hui.1|li.hui.5Consider a Hamiltonian action of the circle on a connected compact symplectic manifold \((M, \omega)\) of dimension \(2n\). Then this circle action has at least \(n + 1\) fixed points. Having previously discussed the case when the fixed point set consists of exactly \(n + 1\) isolated points in [``Hamiltonian circle actions with minimal isolated fixed points'', Preprint, \url{arXiv:1407.1948}], the author discusses in this work the case when the fixed point set consists of exactly \(n+2\) isolated points. The first observation is that in such a case \(n\) must be even.
The main result of this work describes conditions on the image of the fixed point set by the moment map, which imply that the cohomology ring and the total Chern class of \(M\) are isomorphic to that of \(\widetilde G_2(\mathbb P^{n+2})\) and that the sets of weights of the circle action on \(M\) are those of a standard action on \(\widetilde G_2(\mathbb P^{n+2})\).
Reviewer: Elizabeth Gasparim (Antofagasta)Uniqueness of real Lagrangians up to cobordismhttps://www.zbmath.org/1483.530952022-05-16T20:40:13.078697Z"Kim, Joontae"https://www.zbmath.org/authors/?q=ai:kim.joontaeAn antisymplectic involution on a symplectic manifold is an involution which acts on the symplectic structure in an antisymmetric fashion. The nonempty fixed point sets of such involutions are necessarily Lagrangian. If a Lagrangian submanifold is the fixed point set of an antisymplectic involution, it is called real. The main result of the present article is that any two real Lagrangians in a closed symplectic manifold are smoothly cobordant; i.e. they represent the same cobordism class in the nonoriented Thom cobordism ring (see e.g. [\textit{C. T. C. Wall}, Ann. Math. (2) 72, 292--311 (1960; Zbl 0097.38801)] for a complete algebraic description). In case the Lagrangian is not real, this claim need not be true in general. The proof of the main result is through the observation that the cobordism class of the symplectic manifold containing the real Lagrangian \(L\) is equal to the class of \(L \times L\). It follows immediately from the main result that if two real Lagrangians \(L_1\) and \(L_2\) in \(M_1\) and \(M_2\) respectively are cobordant, then \(M_1\) and \(M_2\) are cobordant too.
The main result and various previous results in the literature (e.g. those coming from Smith theory) are brought together smartly in the article to conclude interesting observations regarding real Lagrangians. For example, any real Lagrangian in \(\mathbb{C}P^2\) is Hamiltonian isotopic to \(\mathbb{R}P^2\). As for real Lagrangians in \(S^2\times S^2\), \(L\) is either Hamiltonian isotopic to the antidiagonal sphere, or Lagrangian isotopic to the Clifford torus \(S^1\times S^1\subset S^2\times S^2\). Let us note that the general Lagrangian classification problem is much more complicated. Along that direction the author collects basic examples and diverse counter-examples.
The article is written clearly and is easy-to-read.
Reviewer: Ferit Öztürk (İstanbul)\(S^1\)-invariant symplectic hypersurfaces in dimension 6 and the Fano conditionhttps://www.zbmath.org/1483.531012022-05-16T20:40:13.078697Z"Lindsay, Nicholas"https://www.zbmath.org/authors/?q=ai:lindsay.nicholas"Panov, Dmitri"https://www.zbmath.org/authors/?q=ai:panov.dmitriIn the paper under review the authors study symplectic Fano \(6\)-manifolds equipped with a Hamiltonian \(S^1\)-action. Let \(M\) be such a manifold, and let \(M_{\min}\) be the set of points where a Hamiltonian that generates the action attains its minimal value. It turns out that \(M_{\min}\) is connected. The main result (Theorem~1.3) states that \(M_{\min}\) is diffeomorphic to either a del Pezzo surface, a \(2\)-sphere or a point. By the result of [\textit{H. Li}, Proc. Am. Math. Soc. 131, No. 11, 3579--3582 (2003; Zbl 1066.53127)] the fundamental group of \(M\) is isomorphic to that of \(M_{\min}\), hence, as consequence, \(M\) must be simply connected. The Todd genus is also determined (Corollary~1.4). The proof is split into several cases, depending on the dimensions of \(M_{\min}\) and of the set \(M_{\max}\) where the generating Hamiltonian takes its maximum. The arguments are intricate, and rely on several interesting tools such as the notion of a \textit{symplectic fibre}, whose existence is proved under the assumption that \(M_{\min}\) is two dimensional (Theorem~1.5), and the existence of surfaces of fixed points with restricted topology (Theorem~1.8).
Reviewer: Umberto Leone Hryniewicz (Rio de Janeiro)Tautological rings and stabilisationhttps://www.zbmath.org/1483.550082022-05-16T20:40:13.078697Z"Randal-Williams, Oscar"https://www.zbmath.org/authors/?q=ai:randal-williams.oscarThis paper pertains to the study of certain characteristic classes of smooth manifold bundles known as \textit{tautological classes}. (The words \(\kappa\)-classes and generalized Miller-Morita-Mumford classes are synonyms.) For an oriented connected closed smooth manifold \(M\) of dimension \(d\), let \(\text{Diff}^+(M)\) denote the group of orientation-preserving diffeomorphisms of \(M\), endowed with the Whitney topology, and let \(\text{Diff}^+(M, \star)\) denote the subgroup of diffeomorphisms fixing a basepoint. Then the induced map on classifying spaces, \(\pi \colon B\text{Diff}^+(M, \star) \to B\text{Diff}^+(M)\), is a model of the universal oriented smooth \(M\)-bundle. The map induced from the differential of a diffeomorphism at the basepoint yields a map \(D\colon B\text{Diff}^+(M,\star) \to B\text{GL}^+_d(\mathbb R) \sim BSO(d)\). For any \(c \in H^{\ast}(BSO(d);\mathbb Q)\), the corresponding tautological class is defined as \(\kappa_c = \int_{\pi} D^{\ast}(c)\), where the integral stands for integration along fibers -- a map that shifts cohomological degree down by \(d\). The subring of \(H^{\ast}(B\text{Diff}^+(M);\mathbb Q)\) generated by all such classes is the tautological ring of \(M\), denoted \(R^{\ast}(M)\). The subring of \(H^{\ast}(B\text{Diff}^+(M, \star);\mathbb Q)\) generated by all classes \(\pi^{\ast} \kappa_c\) and \(D^{\ast}c\) is a variant, the tautological ring fixing a point, and is denoted by \(R^{\ast}(M,\star)\).
The main result of the paper is to construct, for \(d = 2(a+b)\) and \(N\) arbitrary, an explicit ring homomorphism \(R^{\ast}(N \# S^{2a}\! \times \!S^{2b}, \star) \to R^{\ast}(N, \star)\). It sends, for any sequence \(I = (i_1, \dots, i_r)\) of integers in \([1,n]\), the tautological class arising from the monomial \(p_I = p_{i_1} \cdots p_{i_r}\) of Pontryagin classes to itself, the class \(\kappa_{ep_i}\) (where \(e\) is the Euler class) to \(\kappa_{ep_i} + 2p_I\), and any class of the form \(D^{\ast}c\) to itself. The author states that this is surprising as there is most likely no corresponding map \(B\text{Diff}^+(N,\star) \to B\text{Diff}(N \# S^{2a} \times S^{2b}, \star)\).
The proof makes use of several explicit geometric constructions like parametrized connected sums and torus actions. To derive cohomological identities, equivariant cohomology is employed. The crucial property of \(S^{2a}\! \times \!S^{2b}\) that the proof relies on is the existence of an action by an \((a+b)\)-dimensional torus with isolated fixed points. It is explained how the method can also be used to stabilize by manifolds other than \(S^{2a} \times S^{2b}\); in particular, the case of \(\mathbb CP^2\) is discussed in detail. Moreover, the main result is applied to several examples, yielding ring-theoretic insights about tautological rings of specific manifolds. In particular, it is deduced that the Krull dimensions of the tautological rings of the manifolds \(W_g^{2n} = (S^{n} \times S^{n})^{\# g}\) constitute a non-decreasing function of \(g = 0,1,2,\dots\) provided that \(n\) is even, whereas for \(n\) odd, this sequence is known to be \(n,0,n-1,n-1,\dots\)
As anything written by Randal-Williams, the paper is illuminating and a pleasure to read.
Reviewer: Jens Reinhold (Münster)Self-referential discs and the light bulb lemmahttps://www.zbmath.org/1483.570182022-05-16T20:40:13.078697Z"Gabai, David"https://www.zbmath.org/authors/?q=ai:gabai.davidThis paper extends the methods of a former paper of the author [J. Am. Math. Soc. 33, No. 3, 609--652 (2020; Zbl 1479.57048)] to discs with applications to knotted 3-balls in 4-manifolds and further questions.
Let \(R\) be an embedded surface with transverse sphere (a geometrically dual sphere) \(G\) in the \(4\)-manifold \(M\) and let \(z = R\cap G\). Let \(\alpha_0\) and \(\alpha_1\) be two smooth compact arcs that coincide near their endpoints and bound the pinched embedded disc \(E\) that is transverse to \(R\) with \(R\cap E=y\) and \(E\cap G=\emptyset\). Then the light bulb lemma (Lemma~2.3 of [loc. cit.]) asserts that one can perform the crossing change of Figure 1 of the paper (and Figure 2.1 of [loc. cit.]) via an isotopy of \(R\) (along \(E\)), provided there is a path \(\alpha\subset R\) from \(y\) to \(z\) that is disjoint from the tube \(B\). Here \(B\) is the intersection of \(R\) with a neighborhood of \(\alpha_0\) in \(M\). The support of the isotopy may be assumed to be within \(B\). The paper ``investigates what happens when such path must cross \(B\), i.e., is self-referential.'' Self-referential forms and self-referential discs are defined in Section 2, and an explanatory example is given in the second paragraph of the first page together with Figure 2.
The paper includes applications, such as Theorem 0.8 and Theorem 5.1, on the existence of knotted \(3\)-balls in \(4\)-manifolds. Theorem 5.1 is as follows.
\textbf{Theorem 5.1} If \(M=S^2\times D^2\natural S^1\times B^3\) and \(\Delta_0= x_0\times B^3\) in the \(S^1\times B^3\) factor, then there exist infinitely many \(3\)-balls properly homotopic to \(\Delta_0\), but not pairwise properly isotopic.
The organization of the paper is described as: ``Basic definitions will be given in Section 1. Section 2 will describe to what extent the methods of [loc. cit.] extend to discs. In particular, we will show that if \(D_0\) and \(D_1\) are homotopic and have a common dual sphere, then \(D_1\) can be put into a self-referential form with respect to \(D_0\). This is the analogue of the normal form of [loc. cit.] except that in addition to double tubes, \(D_1\) can have finitely many self-referential discs. Theorem 0.6 (i) will also be proved. The Dax isomorphism theorem [\textit{J.-P. Dax}, Ann. Sci. Éc. Norm. Supér. (4) 5, 303--377 (1972; Zbl 0251.58003)] will be stated and proved in Section 3. A slightly sharper version of Theorem 0.6 (ii) will be proved in Section 4. Applications to knotted 3-balls in 4-manifolds and further questions will be given in Section 5.''
Reviewer: Ahmet Beyaz (Ankara)Spaces of knotted circles and exotic smooth structureshttps://www.zbmath.org/1483.570192022-05-16T20:40:13.078697Z"Arone, Gregory"https://www.zbmath.org/authors/?q=ai:arone.gregory-z"Szymik, Markus"https://www.zbmath.org/authors/?q=ai:szymik.markusLet \(N_1\) and \(N_2\) be closed smooth manifolds of dimension \(n\). Assume that \(N_1\) and \(N_2\) are homeomorphic. Let \(\mathrm{Emb}({\mathbb{S}}^1,N_1)\) and \(\mathrm{Emb}({\mathbb{S}}^1,N_2)\) denote the spaces of smooth embeddings of the circle into \(N_1\) and \(N_2\), respectively. The authors prove that \(\mathrm{Emb}({\mathbb{S}}^1,N_1)\) and \(\mathrm{Emb}({\mathbb{S}}^1,N_2)\) have the same homotopy \((2n-7)\)-type. Thus the homotopy groups \(\pi_\ast\) of these spaces are isomorphic for \(\ast\leq 2n-7\).
For \(n=4\), the result says that there is a bijective correspondence between \(\pi_0\bigl(\mathrm{Emb}({\mathbb{S}}^1,N_1)\bigr)\) and \(\pi_0(\mathrm{Emb}\bigl({\mathbb{S}}^1,N_2)\bigr)\) and that the corresponding path components have isomorphic fundamental groups \(\pi_1\). The result gives a negative partial answer to a question of Oleg Viro, who asked whether the algebraic topology of the space of smooth \(1\)-knots in a \(4\)-manifold is sensitive to the smooth structure on the ambient manifold.
The proof is based on using manifold functor calculus. The authors give a new model for the quadratic approximation \(\mathrm{T}_2\mathrm{Emb}(M,N)\) of the Goodwillie-Weiss embedding tower, where \(M\) and \(N\) are closed smooth manifolds. They show that the homotopy type of \(\mathrm{T}_2\mathrm{Emb}({\mathbb{S}}^1,N)\) does not depend on the smooth structure of \(N\). They also give a lower bound on \(\pi_2\bigl( \mathrm{Emb}({\mathbb{S}}^1,N)\bigr)\). Moreover, the authors show that for every choice of basepoint, \(\pi_{i}\bigl(\mathrm{Emb}({\mathbb{S}}^1, {\mathbb{S}}^1\times {\mathbb{S}}^3)\bigl)\) contains an infinitely generated free abelian group, where \(i=1,2\).
Reviewer: Marja Kankaanrinta (Helsinki)Taut foliations leafwise branch cover \(S^2\)https://www.zbmath.org/1483.570282022-05-16T20:40:13.078697Z"Calegari, Danny"https://www.zbmath.org/authors/?q=ai:calegari.dannyLet \(M\) be a closed oriented manifold of dimension \(3\), endowed with an oriented, cooriented foliation \(\mathcal F\) of dimension \(2\). \(\mathcal F\) is called taut if there is a compact global transversal to the leaves. Other descriptions of tautness are well known, and a new one is given in the paper: the existence of a map \(\phi:M\to S^2\) whose restriction to every leaf is a branched cover.
The existence of \(\phi\) gives tautness with the following easy argument. After assuming that \(\phi\) is in a general position, where the branching points are of degree 2, we have, either \(M=S^2\times S^1\) foliated by the \(S^2\times\text{point}\), or the branching points form a compact global transversal.
The difficult part is to show the existence of \(\phi\) when \(\mathcal F\) is taut. Two short analytic proofs are given, one using a deep result of \textit{B. Deroin} [J. Inst. Math. Jussieu 7, No. 1, 67--91 (2008; Zbl 1153.32004)], based on previous results of \textit{É. Ghys} [Panor. Synth. 8, 49--95 (1999; Zbl 1018.37028)] and \textit{A. Candel} [Ann. Sci. Éc. Norm. Supér. (4) 26, No. 4, 489--516 (1993; Zbl 0785.57009)], and another one using deep results of \textit{Y. M. Eliashberg} and \textit{W. P. Thurston} [Confoliations. Providence, RI: American Mathematical Society (1998; Zbl 0893.53001)], \textit{J. Bowden} [Geom. Funct. Anal. 26, No. 5, 1255--1296 (2016; Zbl 1362.57037)], \textit{W. H. Kazez} and \textit{R. Roberts} [Geom. Topol. 21, No. 6, 3601--3657 (2017; Zbl 1381.57014)], \textit{Y. Eliashberg} [ibid. 8, 277--293 (2004; Zbl 1067.53070)], \textit{J. B. Etnyre} [Algebr. Geom. Topol. 4, 73--80 (2004; Zbl 1078.53074)] and \textit{S. K. Donaldson} [J. Differ. Geom. 53, No. 2, 205--236 (1999; Zbl 1040.53094)]. The author also provides two additional purely combinatorial proofs, which are explicit and constructive.
The paper also relates invariants of \(\phi\) with invariants of \(\mathcal F\), like the Euler class \(e(T\mathcal F)\), the pull-back \(\phi^*e(S^2)\) of the Euler class of \(S^2\), invariant transverse measures of \(\mathcal F\), and the Hopf invariant of \(\phi\). Finally, the classes that can be realized as \(\phi^*e(S^2)\) are studied.
Reviewer: Jesus A. Álvarez López (Santiago de Compostela)Configuration spaces of squares in a rectanglehttps://www.zbmath.org/1483.570312022-05-16T20:40:13.078697Z"Plachta, Leonid"https://www.zbmath.org/authors/?q=ai:plachta.leonid-pThe author studies the configuration space \(F_k(Q;r)\) of \(k\) squares of size \(r\) in a rectangle \(Q\), by using the tautological function \(\theta\) defined on the affine polytope complex \(Q^k\). The critical points of the function \(\theta\) are described in geometric and combinatorial terms. It is also proved that under certain conditions, the space \(F_k(Q;r)\) is connected. The paper is organized into three sections dealing with the following aspects: Introduction, affine Morse-Bott functions on affine polytope complexes, configuration spaces of squares in a rectangle (first surgery of \(F_k(Q;r)\), connectivity of \(F_n(Q;r)\)).
Reviewer: Dorin Andrica (Riyadh)Flexibility in contact geometry in high dimension [after Borman, Eliashberg and Murphy]https://www.zbmath.org/1483.570322022-05-16T20:40:13.078697Z"Massot, Patrick"https://www.zbmath.org/authors/?q=ai:massot.patrickSummary: Contact structures are hyperplane fields appearing naturally on the boundary of symplectic or holomorphic manifolds, and whose appeal stems from a subtle mixture of rigidity and flexibility. On the rigid side, Gromov's holomorphic curves prove that, in all dimensions, homotopical invariants are not enough to describe deformation classes of contact structures. On the flexible side, which is the topic of this exposition, Borman, Eliashberg and Murphy [\textit{M. S. Borman} et al., Acta Math. 215, No. 2, 281--361 (2015; Zbl 1344.53060)] proved the existence, in all dimensions, of a class of contact structures whose geometry is entirely ruled by algebraic topology. In particular, they give a homotopical characterisation of manifolds carrying contact structures.
For the entire collection see [Zbl 1416.00029].On the topological complexity of Grassmann manifoldshttps://www.zbmath.org/1483.570332022-05-16T20:40:13.078697Z"Ramani, Vimala"https://www.zbmath.org/authors/?q=ai:ramani.vimalaLet \(X\) be a path connected topological space and \(PX\) the space of paths in \(X\). The topological complexity of \(X\), denoted by \(TC(X)\), is the (reduced) Schwarz genus of the fibration \(\pi:PX\rightarrow X\times X\) given by \(\pi(\omega)=(\omega(0),\omega(1))\). The most notable lower bounds for \(TC(X)\) are the \textit{\(K\)-zero-divisor cup-lengths} of \(X\), denoted by \(\mathrm{zcl}_K(X)\), for various fields \(K\). This is the maximal number of elements of positive degree in \(\mathrm{ker}(\cup:H^*(X;K)\otimes H^*(X;K)\rightarrow H^*(X;K))\) with nonzero product in the algebra \(H^*(X;K)\otimes H^*(X;K)\) (where \(\cup\) is the cohomology cup product). Also, a well-known upper bound for topological complexity of an \(r\)-connected polyhedron \(X\) is \(2\dim(X)/(r+1)\).
In this paper the author proves that for any quaternionic flag manifold its topological complexity equals the half of its real dimension. It is shown that the \(\mathbb Q\)-zero-divisor cup-length of such a manifold (a lower bound for the topological complexity) coincides with the above mentioned upper bound.
The other class of manifolds considered in the paper are the Grassmann manifolds \(\widetilde G_{n,k}\) of oriented \(k\)-planes in \(\mathbb R^n\). The author presents a complete calculation of \(\mathrm{zcl}_\mathbb Q(\widetilde G_{n,k})\) (for \(3\leq k\leq [n/2]\)), and in the process completes the work of \textit{J. Korbaš} [Topology Appl. 153, No. 15, 2976--2986 (2006; Zbl 1099.55001)] on computing the \(\mathbb Q\)-cup-length of these manifolds. In some low-dimensional cases the \(\mathbb Z_2\)-zero-divisor cup-length of \(\widetilde G_{n,k}\) is calculated, which turns out to be a better lower bound for \(TC(\widetilde G_{n,k})\) than \(\mathrm{zcl}_\mathbb Q(\widetilde G_{n,k})\).
Reviewer: Branislav Prvulović (Beograd)Euler classes of vector bundles over manifoldshttps://www.zbmath.org/1483.570342022-05-16T20:40:13.078697Z"Naolekar, Aniruddha C."https://www.zbmath.org/authors/?q=ai:naolekar.aniruddha-cThe paper under review is a continuation of the previous paper [\textit{A. C. Naolekar} et al., Homology Homotopy Appl. 22, No. 1, 215--232 (2020; Zbl 1456.57025)]. Let \(\mathcal{E}_k\) be the set of diffeomorphism classes of closed connected smooth \(k\)-manifold \(X\) with the property that the Euler class of every oriented vector bundle over \(X\) is trivial. In the case that the dimension \(k\) is even, the author shows that if \(X\) belongs to \(\mathcal{E}_{k}\), then its Euler characteristic \(\chi (X)\) is non-positive. Moreover, in the case that \(k\) is odd and \(k\geq 5\), some necessary conditions are obtained. That is, if \(X\) is non-orientable, then \(H_{1}(X;\mathbb{Z})\) is infinitely cyclic and \(H_{i}(X;\mathbb{Z})\) is finite for all \(i>1\). If \(X\) is orientable, then \(\pi_{1}(X)\) is perfect, \(X\) is a rational homology \(k\)-sphere, and \(H^{2}(X;\mathbb{Z})=0\). As another restriction in the general setting, the author shows that if \(X\in\mathcal{E}_{k}\) where \(k>5\) is odd, then \(H_{k-4}(X;\mathbb{Z})\) is \(2\)-primary. As a consequence of these results, it is shown that \(\mathcal{E}_{k}\) does not contain products. In the direction of a complete understanding of \(\mathcal{E}_k\) for smaller values of \(k\), the author proves that \(\mathcal{E}_8\) is empty.
Reviewer: Shigeaki Miyoshi (Tokyo)Geometry of surfaces in \(\mathbb{R}^5\) through projections and normal sectionshttps://www.zbmath.org/1483.570352022-05-16T20:40:13.078697Z"Deolindo-Silva, J. L."https://www.zbmath.org/authors/?q=ai:silva.jorge-luiz-deolindo"Sinha, R. Oset"https://www.zbmath.org/authors/?q=ai:sinha.raul-osetThe main focus of the article is the study of geometry of surfaces in \(\mathbb{R}^5\). The first approach used by the authors is to relate the geometry of a surface in \(\mathbb{R}^5\) to that of corresponding (both regular and singular) surfaces in \(\mathbb{R}^4\) obtained by orthogonal projections. In particular, relations between the asymptotic directions of the original surface and those of the projected surface are obtained. It is interesting to note here that the asymptotic directions for surfaces in \(\mathbb{R}^5\), unlike those in \(\mathbb{R}^4\), do not depend only on the second order geometry of the surface.
The authors also establish relations between the umbilical curvatures for surfaces in \(\mathbb{R}^5\) [\textit{S. I. R. Costa} et al., Differ. Geom. Appl. 27, No. 3, 442--454 (2009; Zbl 1176.53015)] and their projected surfaces in \(\mathbb{R}^4\). The authors study the contact between the projected surfaces with spheres in \(\mathbb{R}^4\) and show that there exists a unique umbilical focal hypersphere at a point of the surface if and only if there exists a unique umbilic focal hypersphere at the corresponding point on the projected surface.
Surfaces in \(\mathbb{R}^5\) can also be obtained as normal sections of 3-manifolds in \(\mathbb{R}^6\), so the authors then go on to consider the geometry of surfaces in \(\mathbb{R}^5\) by relating the asymptotic directions at a point in the 3-manifold with asymptotic directions at the corresponding point in the normal section. By introducing an appropriate umbilic curvature for 3-manifolds, they then study the contact with spheres using this invariant and relate it to the contact between spheres and the surface in \(\mathbb{R}^5\) obtained as a normal section.
Reviewer: Graham Reeve (Liverpool)Rigidity of the mod 2 families Seiberg-Witten invariants and topology of families of spin 4-manifoldshttps://www.zbmath.org/1483.570362022-05-16T20:40:13.078697Z"Kato, Tsuyoshi"https://www.zbmath.org/authors/?q=ai:kato.tsuyoshi"Konno, Hokuto"https://www.zbmath.org/authors/?q=ai:konno.hokuto"Nakamura, Nobuhiro"https://www.zbmath.org/authors/?q=ai:nakamura.nobuhiroFrom the introduction: ``In this paper, we study a family version of rigidity results on the \(\mathbb{Z}_2\)-valued Seiberg-Witten invariant. Namely, for a given family of spin 4-manifolds with some topological conditions, we consider the \(\mathbb{Z}_2\)-valued families Seiberg-Witten invariant, and verify that it depends only on weaker information than is a priori expected, cf. [\textit{D. Ruberman} and \textit{S. Strle}, Math. Res. Lett. 7, No. 5--6, 789--799 (2000; Zbl 1005.57016); \textit{S. Bauer}, J. Differ. Geom. 79, No. 1, 25--32 (2008; Zbl 1144.57027)]. Roughly speaking, we verify that the \(\mathbb{Z}_2\)-valued families Seiberg-Witten invariant is determined by the linearization of a family of Seiberg-Witten equations. A mechanism of this rigidity theorem also gives a family version of Furuta's 10/8-inequality [\textit{M. Furuta}, Math. Res. Lett. 8, No. 3, 279--291 (2001; Zbl 0984.57011)] in a suitable situation.''
Let \(M\) be a smooth closed 4-manifold. The article is essentially a study of Seiberg-Witten Theory when one considers a family \(X\) of closed, smooth 4-manifolds as in [\textit{D. Ruberman}, Math. Res. Lett. 5, No. 6, 743--758 (1998; Zbl 0946.57025); Geom. Topol. Monogr. 2, 473--488 (1999; Zbl 0952.57007)]. To define a family, the authors consider \(B\) a closed smooth manifold, \(M\) a closed smooth 4-manifold equipped with a spin structure \(\mathfrak{s}\) and \(M\rightarrow X\rightarrow B\) a fiber bundle whose structure group is \(\mathrm{Homeo}^{+}(M)\), the group of diffeomorphisms preserving orientation.
The family version of the 10/8-type inequality obtained is applied to prove the existence of a non-smoothable family of 4-manifolds. The bundle \(X\) is non-smoothable as a family or has no smooth reduction if for any smooth structure on \(M\) there is no reduction of the structure group of \(X\) to \(\mathrm{Diff}(M)\) via the inclusion \(\mathrm{Diff}(M)\hookrightarrow \mathrm{Homeo}(M)\). Therefore the family \(X\) is non-smoothable if there is no lift of the classifying map \(\phi:B\to\mathrm{BHomeo}(M)\) to a map \(\psi:B\to\mathrm{BDiff}(M)\) along the natural inclusion \(\mathrm{BDiff}(M)\hookrightarrow \mathrm{BHomeo}(M)\) with respect to any smooth structure on \(M\). Indeed, the authors provide the following example: Let \(-E_{8}\) be the definite intersection form and \(K\) the corresponding 4-manifold, and let \(H\) be the intersection form of \(S^2\times S^2\). Consider the 4-manifold \(M\) defined by the intersection form \(M=2(-E_{8})\oplus mH\), where \(3\le m\le6\). For a subset \(I = \{i_1,\dots,i_{k}\}\subset \{1, 2,\dots, m\}\) with cardinality \(k\), denote by \(T^{k}_{I}\) the \(k\)-torus embedded in the \(m\)-torus \(T^{m}=S^1\times\dots\times S^1\) defined as the product of the \(i_1,\dots,i_{k}\) \(S^1\)-components. Then there exists a \(\mathrm{Homeo}(M)\)-bundle \(M\to X\to T^{m}\) over the \(m\)-torus satisfying the following properties:
\begin{itemize}
\item[(i)] The total space \(X\) admits a smooth manifold structure.
\item[(ii)] If \(k\le m-3\), the restricted family \(X|_{T^{k}_{I}}\to T^{k}_{I}\) admits a reduction to \(\mathrm{Diff}(M)\) for some smooth structure on \(M\).
\item[(iii)] If \(m-2\le k\le m\), the restricted family \(X|_{T^{k}_{I}}\to T^{k}_{I}\) has no reduction to \(\mathrm{Diff}(M)\) for any smooth structure on \(M\).
\end{itemize}
Then the example shows that the inclusion \(\mathrm{Diff}(M)\hookrightarrow \mathrm{Homeo}(M)\) is not a weak homotopy equivalence.
As a consequence of the main theorem that the authors prove in \S 4, in \S 5 they give examples of non-smoothable actions on \(M=2(-E_{8})\oplus m H\). In \S 4 they prove the main results in this paper, namely, the rigidity theorem, and its consequences such as the 10/8-type inequality.
To describe the Main Theorem, let us consider the following: Let \(B\) be a closed smooth manifold, \(M\) a closed smooth 4-manifold equipped with a spin structure \(\mathfrak{s}\) and let \(M\to X\to B\) be a fiber bundle whose structure group is \(\mathrm{Diff}_{+}(M)\), the group of diffeomorphisms preserving orientation. Assume that \(X\) admits a fiberwise spin structure \(\mathfrak{s}_{X}\) whose fiber coincides with the given spin structure on \(M\); it is a global spin structure modeled on \(\mathfrak{s}\). In this situation, there are two real vector bundles over \(B\), namely, \(B:H^{+}\to B\) and \(\mathrm{ind}(D)\) (\(D\) Dirac's operator), where the fiber of \(H^{+}\) is \(H^{+}(M)\) which is a maximal dimensional positive-definite subspace of \(H^2(M;\mathbb{R})\) with respect to the intersection form, and \(\mathrm{ind}(D)\) is the virtual Dirac index bundle associated to \(X\to B\). The Dirac operator \(D\) is \(\mathrm{Pin}(2)\)-equivariant since \(D\) is \(\mathbb{H}\)-linear. They define the \(\mathrm{Pin}(2)\)-action on \(H^{+}\) via the surjective homomorphism \(\mathrm{Pin}(2)\to\mathrm{Pin}(2)/ \{\pm 1\}\) and the multiplication by \(\{\pm 1\}\) to real vector spaces. Then \(\mathrm{ind}(D)\) and \(H^{+}\) determine an element in the \(\mathrm{Pin}(2)\)-equivariant \(\mathrm{KO}\)-group:
\[
\alpha = \alpha(X, \mathfrak{s}_{X}):=[\mathrm{ind}(D)] - [H^{+}] \in \mathrm{KO}_{\mathrm{Pin}(2)}(B).
\]
Let \(b^{+}(M) := \dim^{+}(M)\). If \(b^{+}(M) \ge \dim(B)+2\), they define the (mod 2) families Seiberg-Witten invariant \(FSW^{\mathbb{Z}_2}(X, \mathfrak{s}_{X})\in\mathbb{Z}_2\) of \((X, \mathfrak{s}_{X})\). The first main result in this paper claims that \(FSW^{\mathbb{Z}_2}(X, \mathfrak{s}_{X})\) depends only on \(\alpha(X, \mathfrak{s}_{X})\) which is determined by the linearization of a family of Seiberg-Witten equations.
\textbf{Main Theorem}. Let \(M_1\) and \(M_2\) be oriented closed smooth 4-manifolds with spin structures \(\mathfrak{s}_1\) and \(\mathfrak{s}_2\), respectively. Assume the following conditions:
\begin{itemize}
\item[(i)] \(b_1(M_1)=b_1(M_2)=0\), \(b^{+}(M_1) = b^{+}(M_2)\ge \dim(B) +2\);
\item[(ii)] \(-\frac{\mathrm{sign}(M_i)}{4} - 1 - b^{+}(M_i) + \dim(B)=0\); (\(i = 1, 2\)).
\end{itemize}
For \(i = 1, 2\), let \(X_i\to B\) be a smooth fiber bundle whose fiber is \(M_i\) equipped with a global spin structure \(\mathfrak{s}_{X_i}\) modeled on \(\mathfrak{s}_i\). If \(\alpha(X_1, \mathfrak{s}_{X_1})= \alpha(X_1, \mathfrak{s}_{X_1})\) holds in \(\mathrm{KO}_{\mathrm{Pin}(2)}(B)\), then the equality
\[
FSW^{\mathbb{Z}_2}(X_1, \mathfrak{s}_{X_1})=FSW^{\mathbb{Z}_2}(X_2, \mathfrak{s}_{X_2})\tag{1}
\]
holds.
The Theorem reduces the identity given by Eq. (1) to compare the linearized parts \(\alpha(X_i, \mathfrak{s}_{X_i})\), \(i=1,2\), which is far easier to handle than to compute \(FSW^{\mathbb{Z}_2}(X_i, \mathfrak{s}_{X_i})\), \(i=1,2\). All applications follow from the Main Theorem.
Reviewer: Celso M. Doria (Florianapolis)Non-formality in PIN(2)-monopole Floer homologyhttps://www.zbmath.org/1483.570372022-05-16T20:40:13.078697Z"Lin, Francesco"https://www.zbmath.org/authors/?q=ai:lin.francescoIn a sequence of papers, the author defined \(\mathrm{Pin}(2)\)-equivariant monopole Floer homology \(\mathit{HS}_{\bullet}(Y,\mathfrak{s})\), for a closed oriented \(3\)-manifold with self-conjugate \(\mathrm{spin}^c\) structure \(\mathfrak{s}\) and developed its properties. In particular, a chain complex model of \(\mathit{HS}_{\bullet}(Y,\mathfrak{s})\) is an \(A_\infty\)-module over \(\hat{C}^j_*(S^3)\), the \(\mathrm{Pin}(2)\)-Floer homology of \(S^3\), which is itself an \(A_\infty\) algebra.
It is a natural question to ask the behavior of \(\mathit{HS}_{\bullet}\) under connected sum of \(\mathrm{spin}^c\)-three manifolds. It turns out that this question is much more complicated than for monopole Floer homology \(\mathit{HM}_{\bullet}\). Earlier, the author of the present paper had also shown that \(\mathit{HS}(Y_1\#Y_2)\) is an \(A_\infty\)-tensor product of the \(A_\infty\)-modules of the two components.
As an \(A_\infty\)-tensor product, there is an \emph{Eilenberg-Moore} spectral sequence starting at \(\mathrm{Tor}^{H^*(BPin(2))}_{*,*}(\widehat{HS}_{\bullet}(Y_1),\widehat{\mathit{HS}}_\bullet(Y_2))\) and converging to \(\widehat{\mathit{HS}}(Y_1\#Y_2)\). The higher differentials on this spectral sequence are, in practice, very difficult to compute.
The present paper shows that many calculations of \(\mathrm{Pin}(2)\)-equivariant Floer homology are possible using this spectral sequence, however, frequently also taking advantage of the Gysin sequence for monopole Floer homology. Indeed, one of the key observations of the present paper is a calculation of certain Massey products of \(\widehat{\mathit{HS}}_\bullet(Y,\mathfrak{s})\) in terms of only the Gysin sequence. The paper contains a large collection of manifolds for which calculations are completed, and also shows that certain examples, namely the `manifolds of simple type' (of which there are a great many, including many Seifert spaces and some surgeries on \(L\)-space knots), have surprisingly good behavior under connected sum.
Reviewer: Matthew Stoffregen (Cambridge)Polytope Novikov homologyhttps://www.zbmath.org/1483.570382022-05-16T20:40:13.078697Z"Pellegrini, Alessio"https://www.zbmath.org/authors/?q=ai:pellegrini.alessioFor a closed smooth oriented and connected finite dimensional manifold \(M\), Sergey P. Novikov associated a homology with each a cohomology class \(a\in H^1_\mathrm{dR}(M)\), the so-called Novikov homology \(HN_\ast(a)\), cf. [\textit{S. P. Novikov}, Sov. Math., Dokl. 24, 222--226 (1981; Zbl 0505.58011); translation from Dokl. Akad. Nauk SSSR 260, 31--35 (1981), Russ. Math. Surv. 37, No. 5, 1--56 (1982; Zbl 0571.58011); translation from Usp. Mat. Nauk 37, No. 5(227), 3--49 (1982)]. Let \(\Phi_a:\pi_1(M)\to\mathbb{R}\) be the period homomorphism, and let \(\pi:\widetilde{M}_a\to M\) be the minimal regular covering with the group of deck transformations \(\Gamma_a\cong\pi_1(M)/\mathrm{Ker}(\Phi_a)\). Then for any representative \(\alpha\in a\) there exists an \(\tilde{f}_\alpha\in C^\infty(\widetilde{M}_a)\) such that \(\pi^\ast\alpha=d\tilde{f}_\alpha\). For a Riemannian metric \(g\) on \(M\) the pair \((\alpha, g)\) is said to be Morse-Smale if \((\tilde{f}_\alpha, \pi^\ast g)\) satisfies the Morse-Smale condition on \(\widetilde{M}_a\). For each \(i\in\mathbb{N}_0:=\mathbb{N}\cup\{0\}\) let \(\mathrm{Crit}_i(\tilde{f}_\alpha)\) denote the critical points of \(\tilde{f}_\alpha\) with Morse index \(i\). The \(i\)th Novikov chain group \(\mathrm{CN}_i(\alpha)\) consists of all formal sums
\[
\xi=\sum_{\tilde{x}\in \mathrm{Crit}_i(\tilde{f}_\alpha)}\xi_{\tilde{x}}\tilde{x}\in \bigoplus_{\tilde{x}\in \mathrm{Crit}_i(\tilde{f}_\alpha)} \mathbb{Z}\langle\tilde{x}\rangle
\]
such that \(\{\tilde{x}\mid \xi_{\tilde{x}}\in\mathbb{Z}\setminus\{0\}\,\&\, \tilde{f}_\alpha(\tilde{x})>c\}\) is finite for each \(c\in\mathbb{R}\). The boundary operator \(\partial : \mathrm{CN}_i(\alpha) \to \mathrm{CN}_{i-1}(\alpha)\) is defined by
\[
\partial \xi:=\sum_{\tilde{x}, \, \tilde{y}} \, \xi_{\tilde{x}} \cdot \#_{\mathrm{alg}} \, \underline{\mathcal{M}}(\tilde{x},\tilde{y};\tilde{f}_{\alpha}) \, \tilde{y},
\]
where \(\#_{\mathrm{alg}} \, \underline{\mathcal{M}}(\tilde{x},\tilde{y};\tilde{f}_{\alpha})\) counts trajectories of negative gradient of \(\tilde{f}_\alpha\) with respect to \(\tilde{g}:=\pi^\ast g\) with signs from \(\tilde{x}\) to \(\tilde{y}\).
The Novikov ring \(\Lambda_\alpha\) consists of all formal sums
\[
\lambda=\sum_{A\in\Gamma_a}\lambda_A A\in \bigoplus_{A\in\Gamma_a}\mathbb{Z}\langle A\rangle
\]
such that \(\{A\in\Gamma_a\mid \lambda_A\in\mathbb{Z}\setminus\{0\}\,\&\, \Phi_a(A)<c\}\) is finite for each \(c\in\mathbb{R}\). The product is given by the convolution
\[
(\lambda\ast\mu)_A=\sum_{B\in\Gamma_a}\lambda_B\mu_{B^{-1}A}.
\]
According to the obvious action of \(\Lambda_a\) on \(\mathrm{CN}_\ast(\alpha)\), the latter is a finitely generated \(\Lambda_a\)-module. Moreover the boundary operator \(\partial\) is \(\Lambda_a\)-linear, and for each \(i \in \mathbb{N}_0\) the Novikov homology
\[
\mathrm{HN}_i(\alpha,g):=\frac{\ker \partial_i}{\mathrm{im} \, \partial_{i+1}}
\]
carries a \(\Lambda_a\)-module structure. Different choices of cohomologous Morse forms representing \(\alpha\) induce isomorphic Novikov homologies. The isomorphism class is said to be the Novikov homology of pairs \((\alpha, g)\), and denoted by \(\mathrm{HN}_\ast(a)\).
In the paper under review the author generalizes the above Novikov homology and defines polytope Novikov homology. Corresponding to a polytope \(\mathcal{A}=\langle a_0, \dots, a_k \rangle \subset H^1_{\mathrm{dR}}(M)\) with vertices \(a_0,\dots,a_k\), there exists a regular cover \(\pi : \widetilde{M}_{\mathcal{A}} \to M\) with the group of deck transformations
\[ \Gamma_\mathcal{A}\cong {\pi_1(M)}{\bigg /}\bigcap_{l=0}^k \mathrm{Ker}(\Phi_{a_l}), \]
Then for every \(a \in \mathcal{A}\) and for any representative \(\alpha\in a\) there exists a \(\tilde{f}_{\alpha} \in C^{\infty}(\widetilde{M}_{\mathcal{A}})\) such that \(\pi^*\alpha=d\tilde{f}_{\alpha}\). Fix a smooth section \(\theta : \mathcal{A} \longrightarrow \Omega^1(M)\), that is, \(\theta_a\) is a representative of \(a\). For each \(i\in\mathbb{N}_0\) the \(i\)th polytope Novikov chain complex group \(\mathrm{CN}_i(\theta_a,\mathcal{A})\) consists of all formal sums
\[ \xi=\sum_{\tilde{x}\in \mathrm{Crit}_i(\tilde{f}_{\theta_a})}\xi_{\tilde{x}}\tilde{x}\in \bigoplus_{\tilde{x}\in \mathrm{Crit}_i(\tilde{f}_\alpha)} \mathbb{Z}\langle\tilde{x}\rangle \]
such that
\begin{gather*}
\xi=\sum_{\tilde{x} \in \mathrm{Crit}_i\left(\tilde{f}_{\theta_a}\right)} \xi_{\tilde{x}} \, \tilde{x} \in \mathrm{CN}_i(\theta_a, \mathcal{A}) \iff \forall b \in \mathcal{A}, \forall c \in \mathbb{R} : \\ \#\lbrace \tilde{x} \mid \xi_{\tilde{x}} \neq 0, \; \tilde{f}_\beta(\tilde{x})>c \rbrace < +\infty,
\end{gather*}
where \(\beta \in b\) is any representative. The groups \(\mathrm{CN}_\bullet(\theta_a,\mathcal{A})\) may be equipped with boundary operators \(\partial_{\theta_a} : \mathrm{CN}_\ast(\theta_a,\mathcal{A}) \to \mathrm{CN}_{\ast-1}(\theta_a,\mathcal{A})\) given by
\[ \partial_{\theta_a} \xi:= \sum_{\tilde{x}, \tilde{y}} \xi_{\tilde{x}} \cdot \#_{\mathrm{alg}} \, \underline{\mathcal{M}}\left(\tilde{x},\tilde{y};\tilde{f}_{\theta_a}\right) \, \tilde{y}. \]
Let \(\widehat{\mathbb{Z}}[\Gamma_{\mathcal{A}}]^b\) denote the upward completion of the group ring \(\mathbb{Z}[\Gamma_{\mathcal{A}}]\) with respect to the period homomorphism \(\Phi_b : \Gamma_{\mathcal{A}} \to \mathbb{R}\). Define the polytope Novikov ring \(\Lambda_\mathcal{A}=\bigcap_{b \in \mathcal{A}} \widehat{\mathbb{Z}}[\Gamma_{\mathcal{A}}]^b\). The above boundary operator \(\partial_{\theta_a}\) is \(\Lambda_{\mathcal{A}}\)-linear. The homology of the chain complex \(\left(\mathrm{CN}_\ast(\vartheta_a,g_{\vartheta_a},\mathcal{A}),\partial \right)\), denoted by \(\mathrm{HN}_\ast(\vartheta_a,\mathcal{A})\), is called the polytope Novikov homology. It is proved that any two cohomology classes in a prescribed polytope give rise to chain homotopy equivalent polytope Novikov complexes over a Novikov ring associated with said polytope.
An important application is to present a novel approach to the (twisted) Novikov Morse Homology Theorem: For any cohomology class \(a \in H^1_{\mathrm{dR}}(M)\) there exists an isomorphism \(\mathrm{HN}_\ast (a) \cong \mathrm{H}_\ast(M,\Lambda_a)\) of Novikov-modules.
The second application is to prove a new polytope Novikov Principle, which generalizes the ordinary Novikov Principle and a recent result of Pajitnov in the abelian case [\textit{A. Pajitnov}, Eur. J. Math. 6, No. 4, 1303--1341 (2020; Zbl 1470.57050)].
Reviewer: Guang-Cun Lu (Beijing)Singularities of singular solutions of first-order differential equations of clairaut typehttps://www.zbmath.org/1483.580102022-05-16T20:40:13.078697Z"Saji, Kentaro"https://www.zbmath.org/authors/?q=ai:saji.kentaro"Takahashi, Masatomo"https://www.zbmath.org/authors/?q=ai:takahashi.masatomoThe work of this paper is a part of an ongoing research on understanding singularities of envelopes for differential equations of Clairaut type. Let us explain the main notions and concepts. First, consider the ordinary differential equation \[ F(x,y,p)=0, \tag{1} \] where \(p\) stands for the derivative \(dy/dx\) and \(F\) is defined in a domain of the space \(J^1(\mathbb{R},\mathbb{R})\) that consists of 1-jets of functions \(y(x)\), i.e., the space with coordinates \(x,y,p\) equipped with the contact 1-form \(pdx - dy = 0\). Assume that equation (1) defines a smooth surface in \(J^1(\mathbb{R},\mathbb{R})\), then the contact structure cuts the vector field \[ X = F_p \partial_x + pF_p \partial_y + (F_x+pF_y) \partial_p \] on this surface. Integral curves of the field \(X\) are 1-jet extensions of solutions of equation (1). The canonical projection \(\pi(x,y,p) = (x,y)\) restricted to the surface \(\{F=0\}\) has singular points on the set \(\{F=F_p=0\}\) called the criminant, and the projection of the criminant is the discriminant set of equation (1). Generically, the criminant and the discriminant set are curves, the field \(X\) vanishes at isolated points of the criminant, and the discriminant set is the locus of singularities of solution of (1) (almost all of which are 3:2-cusps). However, there exist a special class of equations (1) called Clairaut type. It is defined by the condition that the function \(F_x+pF_y\) vanishes on the criminant identically. Under light additional conditions, in this case the discriminant set is the envelop of solutions of (1), and consequently, it is a solution as well. The basic examples are \(p^2 = y\) and classical Clairaut's equation itself: \[ f(p) = xp-y, \ \ f''(p) \not\equiv 0. \tag{2} \] The discriminant set of (2) is the dual Legendrian curve to the graph \(y=f(x)\), it is the envelop of its tangent lines \(xc-y=f(c)\), \(c=const\), which are also solutions of (2). It is regular at points where \(f''(p) \neq 0\) are it is singular if \(f''(p)=0\). For instance, it has 3:2-cusps at points where \(f''(p)=0\), \(f'''(p) \neq 0\). This is the simplest example of singularities of envelopes.
Second, the authors investigate the partial differential equation \[ F(x_1, x_2,y,p_1, p_2)=0, \tag{3} \] where \(p_i\) stands for the derivative \(dy/dx_i\) and \(F\) is defined in a domain of the space \(J^1(\mathbb{R}^2,\mathbb{R})\) that consists of 1-jets of functions \(y(x_1,x_2)\), i.e., the space with coordinates \(x_1, x_2,y,p_1, p_2\) equipped with the contact 1-form \(p_1 dx_1+ p_2 dx_2 - dy = 0\). Similarly to the above, there exists a special class of equations (3) called Clairaut type. The authors show that (under light additional conditions) Clairaut type equations (3) have envelops of solutions (which are solutions as well) and establish the list of typical singularities of their envelops: cuspidal edge, swallowtail, cuspidal butterfly, cuspidal lips/beaks, etc (frontal singularities).
Third, the authors investigate the system of equations \[ F(x_1, x_2,y,p_1, p_2)=0, \ \ G(x_1, x_2,y,p_1, p_2)=0, \tag{4} \] where the functions \(F, G\) are defined in a domain of the space \(J^1(\mathbb{R}^2,\mathbb{R})\), all notations are similar to (3) and the Poisson bracket \([F,G]\) on the manifold \(\{F=G=0\}\) is identically zero. The authors introduce a natural notion of Clairaut type systems (4) and establish the list of typical singularities of their envelops similar to those for (3).
Reviewer: Alexey O. Remizov (Moskva)The rôle of Coulomb branches in 2D gauge theoryhttps://www.zbmath.org/1483.811072022-05-16T20:40:13.078697Z"Teleman, Constantin"https://www.zbmath.org/authors/?q=ai:teleman.constantinSummary: I give a simple construction of the \textit{Coulomb branches} \(\mathscr{C}_{3,4}(G;E)\) of gauge theory in three and four dimensions, defined by \textit{H. Nakajima} [Adv. Theor. Math. Phys. 20, No. 3, 595--669 (2016; Zbl 1433.81121)] and \textit{A. Braverman} et al. [Adv. Theor. Math. Phys. 22, No. 5, 1071--1147 (2018; Zbl 1479.81043)] for a compact Lie group \(G\) and a polarizable quaternionic representation \(E\). The manifolds \({\mathscr{C}(G;\mathbf{0})}\) are abelian group schemes over the bases of regular adjoint \({G_\mathbb{C}} \)-orbits, respectively conjugacy classes, and \({\mathscr{C}(G;E)}\) is glued together over the base from two copies of \({\mathscr{C}(G;\mathbf{0})}\) shifted by a rational Lagrangian section \({\varepsilon_V} \), representing the Euler class of the \textit{index} bundle of a polarization \({V}\) of \({E} \). Extending the interpretation of \({\mathscr{C}_3(G;\mathbf{0})}\) as ``classifying space'' for topological 2D gauge theories, I characterize functions on \({\mathscr{C}_3(G;E)}\) as operators on the equivariant quantum cohomologies of \({M\times V} \), for compact symplectic \({G}\)-manifolds \({M}\). The non-commutative version has a similar description in terms of the \({\Gamma}\)-class of \({V}\).Microscopic origin of Einstein's field equations and the \textit{raison d'être} for a positive cosmological constanthttps://www.zbmath.org/1483.830082022-05-16T20:40:13.078697Z"Padmanabhan, T."https://www.zbmath.org/authors/?q=ai:padmanabhan.thanu|padmanabhan.t-v"Chakraborty, Sumanta"https://www.zbmath.org/authors/?q=ai:chakraborty.sumantaSummary: In the paradigm of effective field theory, one hierarchically obtains the effective action \(\mathcal{A}_{\mathrm{eff}} [q, \cdots]\) for some low(er) energy degrees of freedom \(q\), by integrating out the high(er) energy degrees of freedom \(\xi\), in a path integral, based on an action \(\mathcal{A} [q, \xi, \cdots]\). We show how one can integrate out a vector field \(v^a\) in an action \(\mathcal{A} [\Gamma, v, \cdots]\) and obtain an effective action \(\mathcal{A}_{\mathrm{eff}}[\Gamma, \cdots]\) which, on variation with respect to the connection \(\Gamma\), leads to the Einstein's field equations and a metric compatible with the connection. The derivation \textit{predicts} a non-zero, positive, cosmological constant, which arises as an integration constant. The Euclidean action \(\mathcal{A} [\Gamma, v, \cdots]\), has an interpretation as the heat density of null surfaces, when translated into the Lorentzian spacetime. The vector field \(v^a\) can be interpreted as the Euclidean analogue of the microscopic degrees of freedom hosted by any null surface. Several implications of this approach are discussed.On a conformal Schwarzschild-de Sitter spacetimehttps://www.zbmath.org/1483.830402022-05-16T20:40:13.078697Z"Culetu, Hristu"https://www.zbmath.org/authors/?q=ai:culetu.hristuSummary: On the basis of the C-metric, we investigate the conformal Schwarzschild - deSitter spacetime and compute the source stress tensor and study its properties, including the energy conditions. Then we analyze its extremal version \((b^2=27m^2\), where \(b\) is the deS radius and \(m\) is the source mass), when the metric is nonstatic. The weak-field
version is investigated in several frames, and the metric becomes flat with the special choice \(b=1/a\), \(a\) being the constant acceleration of the Schwarzschild-like mass or black hole. This form is Rindler's geometry in disguise and is also conformal to a de Sitter metric where the acceleration plays the role of the Hubble constant. In its time dependent version, one finds that the proper acceleration of a static observer is constant everywhere, in contrast with the standard Rindler case. The timelike geodesics along the z-direction are calculated and proves to be hyperbolae.Simpliciality of strongly convex problemshttps://www.zbmath.org/1483.901512022-05-16T20:40:13.078697Z"Hamada, Naoki"https://www.zbmath.org/authors/?q=ai:hamada.naoki"Ichiki, Shunsuke"https://www.zbmath.org/authors/?q=ai:ichiki.shunsukeA multiobjective optimization problem is \(C^r\) simplicial if the Pareto set and the Pareto front are \(C^r\) diffeomorphic to a simplex and, under the \(C^r\) diffeomorphisms, each face of the simplex corresponds to the Pareto set and the Pareto front of a subproblem, where \(r\) is a positive number. The simpliciality is an important property, which can be seen in several practical problems ranging from facility location studied already. The sparse modeling is actively developed now day. If a problem is simplicial, then it is possible to efficiently compute a parametric-surface approximation of the entire Pareto set with few samples points. In this paper, the authors give a specialized transversality theorem on generic linear perturbations of a strongly convex mapping and proof another theorem for a singularity theory to a strongly convex problem. In Section 2, are presented two examples of weakly simplicial problems and some remarks. By lemmas prepared in Section 3, it is proved in Section 4. Moreover, in Section 5, all manifolds are without boundary and assumed to have countable bases. The purpose of this section is to establish the specialized transversality theorem for generically linearly perturbed strongly convex mappings, which is an essential tool for the proof of theorem in Section 6. Section 7 is an appendix, which shows demonstrations of some lemmas used already in previous sections of the article.
Reviewer: Doina Carp (Bucureşti)