scholarly journals On the topological complexity of manifolds with abelian fundamental group

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Daniel C. Cohen ◽  
Lucile Vandembroucq
Keyword(s):  
Fundamental Group ◽  
Large Class ◽  
Closed Manifold ◽  
Topological Complexity ◽  
Nonorientable Surfaces ◽  
Lusternik Schnirelmann ◽  
Complexity Of Manifolds

Abstract We find conditions which ensure that the topological complexity of a closed manifold M with abelian fundamental group is nonmaximal, and see through examples that our conditions are sharp. This generalizes results of Costa and Farber on the topological complexity of spaces with small fundamental group. Relaxing the commutativity condition on the fundamental group, we also generalize results of Dranishnikov on the Lusternik–Schnirelmann category of the cofibre of the diagonal map Δ : M → M × M {\Delta:M\to M\times M} for nonorientable surfaces by establishing the nonmaximality of this invariant for a large class of manifolds.

scholarly journals Triangulations with few vertices of manifolds with non-free fundamental group

2019 ◽  
Vol 149 (6) ◽  
pp. 1453-1463
Author(s):  
Petar Pavešić
Keyword(s):  
Fundamental Group ◽  
Lower Bounds ◽  
Category Theory ◽  
Homology Sphere ◽  
Dimensional Manifold ◽  
Lusternik Schnirelmann ◽  
Few Vertices

AbstractWe study lower bounds for the number of vertices in a PL-triangulation of a given manifold M. While most of the previous estimates are based on the dimension and the connectivity of M, we show that further information can be extracted by studying the structure of the fundamental group of M and applying techniques from the Lusternik-Schnirelmann category theory. In particular, we prove that every PL-triangulation of a d-dimensional manifold (d ⩾ 3) whose fundamental group is not free has at least 3d + 1 vertices. As a corollary, every d-dimensional homology sphere that admits a combinatorial triangulation with less than 3d vertices is PL-homeomorphic to Sd. Another important consequence is that every triangulation with small links of M is combinatorial.

scholarly journals Topological complexity of collision-free motion planning on surfaces

2010 ◽  
Vol 147 (2) ◽  
pp. 649-660 ◽  
Author(s):  
Daniel C. Cohen ◽  
Michael Farber
Keyword(s):  
Motion Planning ◽  
Configuration Space ◽  
Main Tool ◽  
Orientable Surface ◽  
Configuration Spaces ◽  
Topological Complexity ◽  
Algebraic Varieties ◽  
Hodge Structures ◽  
Lusternik Schnirelmann ◽  
Mixed Hodge Structures

AbstractThe topological complexity$\mathsf {TC}(X)$is a numerical homotopy invariant of a topological spaceXwhich is motivated by robotics and is similar in spirit to the classical Lusternik–Schnirelmann category ofX. Given a mechanical system with configuration spaceX, the invariant$\mathsf {TC}(X)$measures the complexity of motion planning algorithms which can be designed for the system. In this paper, we compute the topological complexity of the configuration space ofndistinct ordered points on an orientable surface, for both closed and punctured surfaces. Our main tool is a theorem of B. Totaro describing the cohomology of configuration spaces of algebraic varieties. For configuration spaces of punctured surfaces, this is used in conjunction with techniques from the theory of mixed Hodge structures.

scholarly journals Hopf invariants for sectional category with applications to topological robotics

2019 ◽  
Author(s):  
Jesús González ◽  
Mark Grant ◽  
Lucile Vandembroucq
Keyword(s):  
Topological Complexity ◽  
Cell Complexes ◽  
Lusternik Schnirelmann ◽  
Hopf Invariants

Abstract We develop a theory of generalized Hopf invariants in the setting of sectional category. In particular, we show how Hopf invariants for a product of fibrations can be identified as shuffle joins of Hopf invariants for the factors. Our results are applied to the study of Farber’s topological complexity for two-cell complexes, as well as to the construction of a counterexample to the analogue for topological complexity of Ganea’s conjecture on Lusternik–Schnirelmann category.

scholarly journals On the topological complexity of aspherical spaces

2018 ◽  
Vol 12 (02) ◽  
pp. 293-319 ◽  
Author(s):  
Michael Farber ◽  
Stephan Mescher
Keyword(s):  
Spectral Sequence ◽  
Cohomology Class ◽  
Hyperbolic Group ◽  
Cohomological Dimension ◽  
Topological Complexity ◽  
Canonical Class ◽  
Lusternik Schnirelmann

The well-known theorem of Eilenberg and Ganea [Ann. Math. 65 (1957) 517–518] expresses the Lusternik–Schnirelmann category of an Eilenberg–MacLane space [Formula: see text] as the cohomological dimension of the group [Formula: see text]. In this paper, we study a similar problem of determining algebraically the topological complexity of the Eilenberg–MacLane spaces [Formula: see text]. One of our main results states that in the case when the group [Formula: see text] is hyperbolic in the sense of Gromov, the topological complexity [Formula: see text] either equals or is by one larger than the cohomological dimension of [Formula: see text]. We approach the problem by studying essential cohomology classes, i.e. classes which can be obtained from the powers of the canonical class (as defined by Costa and Farber) via coefficient homomorphisms. We describe a spectral sequence which allows to specify a full set of obstructions for a cohomology class to be essential. In the case of a hyperbolic group, we establish a vanishing property of this spectral sequence which leads to the main result.

On Wh3 of a Bieberbach group

1984 ◽  
Vol 95 (1) ◽  
pp. 55-60
Author(s):  
Andrew J. Nicas
Keyword(s):  
Fundamental Group ◽  
Universal Covering ◽  
Closed Manifold ◽  
Covering Space ◽  
Whitehead Group ◽  
Bieberbach Group ◽  
Universal Covering Space ◽  
Aspherical Manifold

A closed aspherical manifold is a closed manifold whose universal covering space is contractible. There is the following conjecture concerning the algebraic K-theory of such manifolds:Conjecture. Let Γ be the fundamental group of a closed aspherical manifold. Then Whi(Γ) = 0 for i ≥ 0 where Whi(Γ) is the i-th higher Whitehead group of Γ.

scholarly journals A combination theorem for affine tree-free groups

2016 ◽  
Vol 26 (07) ◽  
pp. 1283-1321
Author(s):  
Shane O. Rourke
Keyword(s):  
Abelian Group ◽  
Fundamental Group ◽  
Large Class ◽  
Recent Work ◽  
Isometric Action ◽  
Graph Of Groups ◽  
Finitely Generated Group ◽  
Affine Action ◽  
Relatively Hyperbolic ◽  
Solvable Word

Let [Formula: see text] be an ordered abelian group. We show how a group admitting a free affine action without inversions on a [Formula: see text]-tree admits a natural graph of groups decomposition, where vertex groups inherit actions on [Formula: see text]-trees. We introduce a stronger condition (essential freeness) on an affine action and apply recent work of various authors to deduce that a finitely generated group admitting an essentially free affine action on a [Formula: see text]-tree is relatively hyperbolic with nilpotent parabolics, is locally relatively quasiconvex, and has solvable word, conjugacy and isomorphism problems. Conversely, given a graph of groups satisfying certain conditions, we show how an affine action of its fundamental group can be constructed. Specialising to the case of free affine actions, we obtain a large class of groups that have a free affine action on a [Formula: see text]-tree but that do not act freely by isometries on any [Formula: see text]-tree. We also give an example of a group that admits a free isometric action on a [Formula: see text]-tree but which is not residually nilpotent.

scholarly journals Homotopic distance between maps

2021 ◽  
pp. 1-21
Author(s):  
E. MACÍAS–VIRGÓS ◽  
D. MOSQUERA–LOIS
Keyword(s):  
General Notion ◽  
Topological Complexity ◽  
Lusternik Schnirelmann

Abstract We show that well-known invariants like Lusternik–Schnirelmann category and topological complexity are particular cases of a more general notion, that we call homotopic distance between two maps. As a consequence, several properties of those invariants can be proved in a unified way and new results arise.

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