Special Issue "Algebraic and Geometric Topology"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 October 2015).

Special Issue Editor

Prof. Dr. Yuli B. Rudyak
Website
Guest Editor
Department of Mathematics, University of Florida, 358 Little Hall, PO Box 118105, Gainesville, FL 32611-8105, USA
Interests: algebraic and geometric topology
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Special Issue Information

Daer Colleagues,

Algebraic Topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphic or homotopical equivalence. In greater detail, in considering a functor from topological spaces for certain algebraic objects (groups, rings, etc.), you know that the non-isomorphic groups come from non-homeomorphic spaces. Thus, given two topological spaces, you can try to find enough functors that can distinguish the spaces.

So, on one hand, algebraic topology studies algebraic invariants (functors) in and of themselves. Here, we concern ourselves with homotopical and homological groups, variety (co)bordism theories, K-theories, and other generalized (co)homology theories, etc., and the task is to evaluate/estimate the algebraic invariants of topological spaces, as well as the relations between these invariants. On the other hand, we observe applications of algebraic topology to several extremals problems (i.e., Morse theory, Lusterink--Schnirelmann theory, isoparametric problems, and systolic inequalities), dynamical systems and symplectic topology, variety aspects of fixed points theories, etc. Also, recently, we  have observed applications of algebraic topology to data analysis, robotics, brain research, biochemistry, etc.

Yuli Rudyak
Guest Editor

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Keywords

  • category weight
  • cobordism
  • generalized cohomology theories
  • Lusternik--Schnirelmann category
  • Massey products
  • Svarc genus

Published Papers (2 papers)

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Research

Open AccessArticle
Dynamics and the Cohomology of Measured Laminations
Mathematics 2016, 4(1), 18; https://doi.org/10.3390/math4010018 - 15 Mar 2016
Abstract
In this paper, the interconnection between the cohomology of measured group actions and the cohomology of measured laminations is explored, the latter being a generalization of the former for the case of discrete group actions and cocycles evaluated on abelian groups. This relation [...] Read more.
In this paper, the interconnection between the cohomology of measured group actions and the cohomology of measured laminations is explored, the latter being a generalization of the former for the case of discrete group actions and cocycles evaluated on abelian groups. This relation gives a rich interplay between these concepts. Several results can be adapted to this setting—for instance, Zimmer’s reduction of the coefficient group of bounded cocycles or Fustenberg’s cohomological obstruction for extending the ergodicity \(\mathbb{Z}\)-action to a skew product relative to an \(S^{1}\) evaluated cocycle. Another way to think about foliated cocycles is also shown, and a particular application is the characterization of the existence of certain classes of invariant measures for smooth foliations in terms of the \(L^{\infty}\)-cohomology class of the infinitesimal holonomy. Full article
(This article belongs to the Special Issue Algebraic and Geometric Topology)
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Open AccessArticle
Multiplicative Expression for the Coefficient in Fermionic 3–3 Relation
Mathematics 2016, 4(1), 3; https://doi.org/10.3390/math4010003 - 20 Jan 2016
Cited by 2
Abstract
Recently, a family of fermionic relations were discovered corresponding to Pachner move 3–3 and parameterized by complex-valued 2-cocycles, where the weight of a pentachoron (4-simplex) is a Grassmann–Gaussian exponent. Here, the proportionality coefficient between Berezin integrals in the l.h.s. and r.h.s. of such [...] Read more.
Recently, a family of fermionic relations were discovered corresponding to Pachner move 3–3 and parameterized by complex-valued 2-cocycles, where the weight of a pentachoron (4-simplex) is a Grassmann–Gaussian exponent. Here, the proportionality coefficient between Berezin integrals in the l.h.s. and r.h.s. of such relations is written in a form multiplicative over simplices. Full article
(This article belongs to the Special Issue Algebraic and Geometric Topology)
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