Recent zbMATH articles in MSC 55Rhttps://www.zbmath.org/atom/cc/55R2021-04-16T16:22:00+00:00WerkzeugMarao, about Hopf fibration.https://www.zbmath.org/1456.550082021-04-16T16:22:00+00:00"Berishvili, Guram"https://www.zbmath.org/authors/?q=ai:berishvili.guramSummary: A marao is a cover of a vector space by a set of equidimensional subspaces with pairwise trivial intersections. Such structures give rise to fibrations of particular kind. Naturally occurring examples are described. In particular, it is explained how the classical Hopf fibrations can be uniformly obtained from maraos.A counterexample to Las Vergnas' strong map conjecture on realizable oriented matroids.https://www.zbmath.org/1456.520292021-04-16T16:22:00+00:00"Wu, Pei"https://www.zbmath.org/authors/?q=ai:wu.peiSummary: The Las Vergnas strong map conjecture asserts that any strong map of oriented matroids \(f : \mathcal{M}_1 \rightarrow \mathcal{M}_2\) can be factored into extensions and contractions. This conjecture is known to be false due to a construction by Richter-Gebert, who finds a strong map that is not factorizable; however, in his example, \(\mathcal{M}_1\) is not realizable. The question of whether there exists a non-factorizable strong map between realizable oriented matroids remains open. In this paper, we provide a counterexample to the strong map conjecture on realizable oriented matroids, which is a strong map \(f : \mathcal{M}_1 \rightarrow \mathcal{M}_2\), where \(\mathcal{M}_1\) is an alternating oriented matroid of rank 4 and \(f\) has corank 2. We prove that the map is not factorizable by showing that there is no uniform oriented matroid \(\mathcal{M}^{\prime}\) of rank 3 such that \(\mathcal{M}_1 \rightarrow \mathcal{M}^{\prime} \rightarrow \mathcal{M}_2\).Differential geometry and Lie groups. A second course.https://www.zbmath.org/1456.530012021-04-16T16:22:00+00:00"Gallier, Jean"https://www.zbmath.org/authors/?q=ai:gallier.jean-h"Quaintance, Jocelyn"https://www.zbmath.org/authors/?q=ai:quaintance.jocelynThis book is written as a second course on differential geometry. So the reader is supposed to be familiar with some themes from the first course on differential geometry -- the theory of manifolds and some elements of Riemannian geometry.
In the first two chapters here some topics from linear algebra are provided -- a detailed exposition of tensor algebra and symmetric algebra, exterior tensor products and exterior algebra. These chapters may be useful when studying the material of this book for those students, who did not study these topics in their algebraic course.
Some themes, which are covered in this book, are rather standard for books on differential geometry - they are differential forms, de Rham cohomology, integration on manifolds, connections and curvature in vector bundles, fibre bundles, principal bundles and metrics on bundles. But a number of topics discussed in this book are not always included in courses on differential geometry and are rarely contained in textbooks on differential geometry. The presence of these topics makes this book especially interesting for modern students. Here is a list of some such topics: an introduction to Pontrjagin
classes, Chern classes, and the Euler class, distributions and the Frobenius theorem. Three chapters need to be highlighted separately. Chapter 7 -- spherical harmonics and an introduction to the representations of compact Lie groups. Chapter 8 -- operators on Riemannian manifolds: Hodge Laplacian, Laplace-Beltrami Laplacian, Bochner
Laplacian. Chapter 11 -- Clifford algebras and groups, groups Pin\((n)\), Spin\((n)\).
Not all statements in this book are given with proofs, for some only links to other textbooks are given. But the most important results are given here with complete proofs and accompanied by examples. Each chapter of this book ends with a list of interesting and sometimes very important problems. At the end of the book there is a very detailed list of the notation used (symbol index) and a detailed list (index) of the terms used.
Reviewer: V. V. Gorbatsevich (Moskva)Discrete noncommutative Gel'fand Naĭmark duality.https://www.zbmath.org/1456.460612021-04-16T16:22:00+00:00"Bertozzini, Paolo"https://www.zbmath.org/authors/?q=ai:bertozzini.paolo"Conti, Roberto"https://www.zbmath.org/authors/?q=ai:conti.roberto.1"Pitiwan, Natee"https://www.zbmath.org/authors/?q=ai:pitiwan.nateeSummary: We present, in a simplified setting, a non-commutative version of the well-known Gel'fand-Naĭmark duality (between the categories of compact Hausdorff topological spaces and commutative unital \(C^*\)-algebras), where ``geometric spectra'' consist of suitable finite bundles of one-dimensional \(C^*\)-categories equipped with a transition amplitude structure satisfying saturation conditions. Although this discrete duality actually describes the trivial case of finite-dimensional \(C^*\)-algebras, the structures are here developed at a level of generality adequate for the formulation of a general topological/uniform Gel'fand-Naĭmark duality, fully addressed in a companion work.Topology change of level sets in Morse theory.https://www.zbmath.org/1456.370612021-04-16T16:22:00+00:00"Knauf, Andreas"https://www.zbmath.org/authors/?q=ai:knauf.andreas"Martynchuk, Nikolay"https://www.zbmath.org/authors/?q=ai:martynchuk.nikolayThis paper considers Morse functions \(f \in C^2(M, \mathbb{R})\) on a smooth \(m\)-dimensional manifold without boundary. For such functions and for every critical point \(x \in M\) of \(f\), the Hessian of \(f\) at \(x\) is nondegenerate.
The authors are interested in the topology of the level sets \(f^{-1}(a) = \partial{M^a}\) where \(M^a = \{ x \in M : f(x) \le a\}\) is a so-called sublevel set. The authors address the question of under what conditions the topology of \(\partial{M^a}\) can change when the function \(f\) passes a critical level with one or more critical points. ``Change in topology'' here means roughly that if \(a\) and \(b\) are regular values with \(a < b\), then \(H_k(\partial{M^a};G) \ne H_k(\partial{M^b};G)\) where \(k\) is an integer and the \(H_k\) are homology groups over some abelian group \(G\) when the function \(f\) passes through a level with one or more critical points. A motivation for this question derives from the \(n\)-body problem: does the topology of the integral manifolds always change when passing through a bifurcation level?
The first part of the paper considers level sets of abstract Morse functions that satisfy the Palais-Smale condition and whose level sets have finitely generated homology groups. They show that for such functions the topology of \(f^{-1}(a)\) changes when passing a single critical point if the index of the critical point is different from \(m/2\) where \(m\) is the dimension of the manifold \(M\). Then they move on to consider the case where \(M\) is a vector bundle of rank \(n\) over a manifold \(N\) of dimension \(n\), and where (up to a translation) \(f\) is a Morse function that is the sum of a positive definite quadratic form on the fibers and a potential function that is constant on the fibers. Here the authors have in mind Hamiltonian functions \(H\) on the cotangent bundle of the base manifold \(N\). In this case they show that the topology of \(H^{-1}(h)\) always changes when passing a single critical point if the Euler characteristic of \(N\) is not \(\pm 1\).
Finally the authors apply their results to examples from Hamiltonian and celestial mechanics, with emphasis on the planar three-body problem. There they show that the topology always changes for the planar three-body problem provided that the reduced Hamiltonian is a Morse function with at most two critical points on each level set.
Reviewer: William J. Satzer Jr. (St. Paul)Classifying spaces for families of subgroups for systolic groups.https://www.zbmath.org/1456.200472021-04-16T16:22:00+00:00"Osajda, Damian"https://www.zbmath.org/authors/?q=ai:osajda.damian"Prytuła, Tomasz"https://www.zbmath.org/authors/?q=ai:prytula.tomaszSummary: We determine the large scale geometry of the minimal displacement set of a hyperbolic isometry of a systolic complex. As a consequence, we describe the centraliser of such an isometry in a systolic group. Using these results, we construct a low-dimensional classifying space for the family of virtually cyclic subgroups of a group acting properly on a systolic complex. Its dimension coincides with the topological dimension of the complex if the latter is at least four. We show that graphical small cancellation complexes are classifying spaces for proper actions and that the groups acting on them properly admitthree-dimensional classifying spaces with virtually cyclic stabilisers. This is achieved by constructing a systolic complex equivariantly homotopy equivalent to a graphical small cancellation complex. On the way we develop a systematic approach to graphical small cancellation complexes. Finally, we construct low-dimensional models for the family of virtually abelian subgroups for systolic, graphical small cancellation, and some CAT(0) groups.On LS-category and topological complexity of some fiber bundles and Dold manifolds.https://www.zbmath.org/1456.550022021-04-16T16:22:00+00:00"Naskar, Bikramaditya"https://www.zbmath.org/authors/?q=ai:naskar.bikramaditya"Sarkar, Soumen"https://www.zbmath.org/authors/?q=ai:sarkar.soumenThe Lusternik-Schnirelmann category (or just category)
\(\mbox{cat}(X)\) of a space \(X\) is a classical important numerical homotopy invariant not only in algebraic topology but also in differential topology and dynamical systems. Another important, but more recent, numerical homotopy invariant is the topological complexity \(\mbox{TC}(X)\) of a space \(X\), which was introduced by \textit{M. Farber} [Discrete Comput. Geom. 29 No. 2, 211--221 (2003; Zbl 1038.68130)] in order to study the motion planning problem in robotics from a topological perspective. Under mild conditions, both invariants are known to satisfy the product inequality:
\(\mbox{cat}(X\times Y)\leq \mbox{cat}(X)+\mbox{cat}(Y)-1\)
and \(\mbox{TC}(X\times Y)\leq \mbox{TC}(X)+\mbox{TC}(Y)-1\), respectively.
Starting from the fact that the product space is a trivial bundle, the authors of the paper under review face the problem of extending the above inequalities to some
classes of more general fiber bundles. In this sense they prove that under certain conditions for a fiber bundle \(F\rightarrow E\stackrel{p}{\rightarrow }B\) the inequality
\(\mbox{cat}(E)\leq \mbox{cat}(B)+\mbox{cat}(F)-1\) holds. A similar formula is also given for topological complexity. Then, taking into account these results and considering usual cohomological lower bounds they are able to give, on the one hand, the computation of the category of Dold manifolds; on the other, they establish tight bounds for the topological complexity of a wide class of Dold manifolds.
Finally, some discussion about the equivariant topological complexity of some spaces related to Dold manifolds is also given.
Reviewer: José Calcines (La Laguna)