arXiv research

Paper introduces fat CW complexes including all closed manifolds.

We show a Whitney Approximation Theorem for a continuous map from a manifold to a smooth CW complex. This enables us to show that a topological CW complex is homotopy equivalent to a smooth CW complex in a category of topological spaces. It is also shown that, for any open covering of a smooth CW complex, there exists …

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

Locally flat submanifolds have finite CW complex complements.

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…

We are concerned with bifurcation analysis and control of nonlinear Eulerian flows with non-resonant n-tuple Hopf singularity. The analysis is involved with CW complex bifurcations of flow-invariant Clifford hypertori, where we refer to these toral manifolds by toral CW complexes. We observe from primary to tertiary fl…

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…

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…

The paper introduces optimal transport kernels for comparing cell complexes.

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

New calculations of topological complexity for symplectic CW-complexes.

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…

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

Simplified proofs for splitting homotopy idempotents.

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…

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…

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 introduce a novel combinatorial method to study $Q^{**}$-transformations of group presentations or, equivalently, 3-deformations of CW-complexes of dimension 2. Our procedure is based on a refinement of discrete Morse theory that gives a Whitehead simple homotopy equivalence from a regular CW-complex to the simplifi…

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.

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

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.

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

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…

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 $K(π_{1}(X),1)$.

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 show that a rank two finite group G admits a finite G-CW-complex X homotopy equivalent to a sphere, with rank one prime power isotropy, if and only if G does not p'-involve Qd(p) for any odd prime p. This follows from a more general theorem which allows us to construct a finite G-CW-complex by gluing together a give…

Self-affine tiles homeomorphic to a ball proven for a specific digit set.

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…

The paper is devoted to generalizations of Cencelj-Dranishnikov theorems relating extension properties of nilpotent CW complexes to its homology groups. Here are the main results of the paper: \par {\bf Theorem}. Suppose $L$ is a nilpotent CW complex and $F$ is the homotopy fiber of the inclusion $i$ of $L$ into its in…

The paper classifies Poincaré complexes as topological manifolds.

This is the second of a series of papers which are devoted to a comprehensive theory of maps between orbifolds. In this paper, we develop a basic machinery for studying homotopy classes of such maps. It contains two parts: (1) the construction of a set of algebraic invariants -- the homotopy groups, and (2) an analog o…

If X is a CW complex, one can assign to each point of X an ordered abelian group of finite rank whose subset of positive elements depends continuously on the points of X. A locally trivial bundle which arises in this way we denote by E(X). In the present work we establish a topological classification of such bundles in…

We construct finite volume hyperbolic manifolds with large symmetry groups. The construction makes use of the presentations of finite Coxeter groups provided by Barot and Marsh and involves mutations of quivers and diagrams defined in the theory of cluster algebras. We generalize our construction by assigning to every …

This article introduces vortex nerve complexes in CW (Closure finite Weak) topological spaces, which first appeared in works by P. Alexandroff, H. Hopf and J.H.C. Whitehead during the 1930s. A vortex nerve is a CW complex containing one or more intersecting path-connected cycles. Each vortex nerve has its own distincti…

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…

We study the existence of Riemannian metrics with zero topological entropy on a closed manifold M with infinite fundamental group. We show that such a metric does not exist if there is a finite simply connected CW complex which maps to M in such a way that the rank of the map induced in the pointed loop space homology …

This paper proves a conjecture of Fomin and Shapiro that their combinatorial model for any Bruhat interval is a regular CW complex which is homeomorphic to a ball. The model consists of a stratified space which may be regarded as the link of an open cell intersected with a larger closed cell, all within the totally non…

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.

Algorithm calculates $L^2$-Euler characteristic for complex spaces.

This paper constructs a CW complex homotopy equivalent to spaces of locally convex curves.

We prove a homological version of a conjecture about the homotopy type of diffeomorphism spaces of reducible 3-manifolds.

The main theorem is that if K is a finite CW complex with finite fundamental group G and universal cover homotopy equivalent to a product of spheres X, then G acts smoothly and freely on X x S^n for any n greater than or equal to the dimension of X. If the G-action on the universal cover of K is homologically trivial t…

The paper characterizes stable cohomotopy groups in codimensions two and three, linking algebraic and geometric perspectives.

Rep-tiles fill cubes in any dimension.