arXiv research

Generalizes theorem for topological $G$-manifolds with linear Lie groups $G$.

A countable CW complex $K$ is quasi-finite (as defined by A.Karasev) if for every finite subcomplex $M$ of $K$ there is a finite subcomplex $e(M)$ such that any map $f:A\to M$, where $A$ is closed in a separable metric space $X$ satisfying $XτK$, has an extension $g:X\to e(M)$. Levin's results imply that none of the Ei…

The simplest condition characterizing quasi-finite CW complexes $K$ is the implication $Xτ_h K\implies β(X)τK$ for all paracompact spaces $X$. Here are the main results of the paper: Theorem: If $\{K_s\}_{s\in S}$ is a family of pointed quasi-finite complexes, then their wedge $\bigvee\limits_{s\in S}K_s$ is quasi-fini…

Generalizes manifold results for Lie groups, proving equivariant homotopy type.

Let $G$ be a compact Lie group. (Compact) topological $G$-manifolds have the $G$-homotopy type of (finite-dimensional) countable $G$-CW complexes (2.5). This partly generalizes Elfving's theorem for locally linear $G$-manifolds [Elf96], wherein the Lie group $G$ is linear (such as compact).

Study of one-dimensional non-Hausdorff manifolds and their quotient to CW complexes.

Let L be a countable and locally finite CW complex. Suppose that the class of all metrizable compacta of extension dimension not greater than L contains a universal element which is an absolute extensor in dimension L. Our main result shows that L is quasi-finite.

This paper defines ribbons and ribbon complexes in CW spaces and analyzes their topological properties.

We present an approach to cohomological dimension theory based on infinite symmetric products and on the general theory of dimension called the extension dimension. The notion of the extension dimension $\ExD(X)$ was introduced by A.N.Dranishnikov \cite {D$_5$} in the context of compact spaces and CW complexes. This pa…

Injectivity proven for measure homology of certain wild spaces.

In this note we introduce the concept of a quasi-finite complex. Next, we show that for a given countable and locally finite CW complex L the following conditions are equivalent: (i) L is quasi-finite. (ii) There exists a [L]-invertible mapping of a metrizable compactum X with e-dim X = [L] onto the Hilbert cube. Final…

Paper introduces fat CW complexes including all closed manifolds.

Kropholler's class of groups is the smallest class of groups which contains all finite groups and is closed under the following operator: whenever $G$ admits a finite-dimensional contractible $G$-CW-complex in which all stabilizer groups are in the class, then $G$ is itself in the class. Kropholler's class admits a hie…

The paper shows how to approximate continuous maps to smooth CW complexes.

Survey on finite group actions on CW-complexes homotopy to spheres.

Locally flat submanifolds have finite CW complex complements.

Study shows $S^1$ algebraic structure in 2-dimensional CW-complex cobordisms.

We show that the Prüfer surface, which is a separable non-metrizable 2-manifold, has not the homotopy type of a CW-complex. This will follow easily from J. H. C. Whitehead's result: if one has a good approximation of an arbitrary space by a CW-complex, which fails to be a homotopy equivalence, then the given space is n…

The paper introduces optimal transport kernels for comparing cell complexes.

Study controls bifurcations in Eulerian flows with multiple Hopf singularities.

New calculations of topological complexity for symplectic CW-complexes.

Euler's theorem extended to complex structures.

We investigate one-point reduction methods of finite topological spaces. These methods allow one to study homotopy theory of cell complexes by means of elementary moves of their finite models. We also introduce the notion of h-regular CW-complex, generalizing the concept of regular CW-complex, and prove that the h-regu…

Minimal example found for two finite CW-complexes sharing a common covering.

It is proved that every discrete Morse function in the sense of Forman on a finite regular CW complex can be represented by a polyhedral Morse function in the sense of Banchoff on an appropriate embedding in Euclidean space of the barycentric subdivision of the CW complex; such a representation preserves critical point…

New method refines Morse theory for group presentations.

The paper introduces vortex nerve complexes and new Betti numbers in CW spaces.

The article defines hyperconnected relator spaces and their properties.

Totally nonnegative flag varieties are shown to be regular CW complexes.

We expand Topological Field Theory on some special CW-complexes (brane complexes). This Brane Topological Field Theory one-to-one corresponds to infinite dimensional Frobenius Algebras, graduated by CW-complexes of lesser dimension. We define general and regular Hurwitz numbers of brane complexes and prove that they ge…

We consider an embedding of a $2$-dimensional CW complex into the $3$-sphere, and construct it's dual graph. Then we obtain a homogeneous system of linear equations from the $2$-dimensional CW complex in the first homology group of the complement of the dual graph. By checking that the homogeneous system of linear equa…

Two complexes share a common covering but not a finite one.

Simplified proofs for splitting homotopy idempotents.

Researchers show a complex structure is not a counterexample to a topological problem.

Random walks on cell complexes link to Laplacians and Novikov-Shubin invariants.

Let $X$ be a finite aspherical CW-complex whose fundamental group $π_1(X)$ possesses a subnormal series $π_1(X) \rhd G_m \rhd ... \rhd G_0$ with a non-trivial elementary amenable group $G_0$. We investigate the $L^2$-invariants of the universal covering of such a CW-complex $X$. We show that the Novikov-Shubin invarian…

We prove that if $M$ is a CW-complex, then the homotopy type of the skeletal filtration of $M$ does not depend on the cell decomposition of $M$ up to wedge products with $n$-disks $D^n$, when the later are given their natural CW-decomposition with unique cells of order 0, $(n-1)$ and $n$; a result resembling J.H.C. Whi…

For any compact Lie group G we discuss the relation of the equivariant Reidemeister and analytic torsion of G-manifolds with their G-CW structures.

Groups with special properties always have fixed points.

We derive a discrete analogue of Morse-Bott theory on CW complexes and use this discrete Morse-Bott function to do some Conley theory analysis. It turns out that our discrete Morse-Bott theory is indeed a generalization of Forman's discrete Morse theory.

We define what it means for a proper continuous morphism between groupoids to be Haar system preserving, and show that such a morphism induces (via pullback) a *-morphism between the corresponding convolution algebras. We proceed to provide a plethora of examples of Haar system preserving morphisms and discuss connecti…

This article introduces descriptive cellular homology on cell complexes, which is an extension of J.H.C. Whitehead's CW topology. A main result is that a descriptive cellular complex is a topology on fibres in a fibre bundle. An application of two forms of cellular homology is given in terms of the persistence of shape…

The paper classifies Poincaré complexes as topological manifolds.

Let X be a finite CW-complex of dimension q. If its fundamental group $π_{1}(X)$ is polycyclic of Hirsch number h>q we show that at least one of the homotopy groups $π_{i}(X)$ is not finitely generated. If h=q or h=q-1 the same conclusion holds unless X is an Eilenberg-McLane space $K(π_{1}(X),1)$.

The paper studies twisted Morse homology and cohomology on manifolds.

Let G be a rank two finite group, and let $\cH$ denote the family of rank one p-subgroups of G, at all primes where G has p-rank two. We show that a rank two finite group G which satisfies certain group-theoretic conditions admits a finite G-CW-complex X with isotropy in $\cH$, whose fixed sets are homotopy spheres. Ou…

The approach we present is a modification of the Morse theory for unital C*-algebras. We provide tools for the geometric interpretation of noncommutative CW complexes. These objects were introduced and studied in [2],[7] and [14]. Some examples to illustrate these geometric information in practice are given. A classifi…

We present a discrete Morse-theoretic method for proving that a regular CW complex is homeomorphic to a sphere. We use this method to define bisimplices, the cells of a class of regular CW complexes we call bisimplicial complexes. The 1-skeleta of bisimplices are complete bipartite graphs making them suitable in constr…