Related papers: When are two spaces homotopy equivalent?

We prove that the classifying space of a simplicial group is modeled by its homotopy coherent nerve.

The homology groups of a simplicial complex reveal fundamental properties of the topology of the data or the system and the notion of topological stability naturally poses an important yet not fully investigated question. In the current…

Homotopy links have proven to be one of the most powerful tools of stratified homotopy theory. In previous work, we described combinatorial models for the generalized homotopy links of a stratified simplicial set. For many purposes, in…

In this article, we construct a cofibrantly generated model structure on the category of spaces stratified over a fixed poset, and show that it is Quillen-equivalent to a category of diagrams of simplicial sets. Then, considering all those…

The matching complex of a simple graph $G$ is a simplicial complex consisting of the matchings on $G$. Jeli\'c Milutinovi\'c et al. studied the matching complexes of the polygonal line tilings, and they gave a lower bound for the…

This paper aims to help the development of new models of homotopy type theory, in particular with models that are based on realizability toposes. For this purpose it develops the foundations of an internal simplicial homotopy that does not…

For a system of non-homogeneous polynomials it was constructed explicit complex morphism of a dual complex to the Koszul complex into the Koszul complex. If the ideal of these polynomials is 0-dimensional, then this mapping is a homotopic…

Let $X$ be a finite CW complex and let $h_1, h_2: C(X)\to A$ be two unital \hm s, where $A$ is a unital C*-algebra. We study the problem when $h_1$ and $h_2$ are approximately homotopic. We present a $K$-theoretical necessary and sufficient…

It is an old conjecture, that finite $H$-spaces are homotopy equivalent to manifolds. Here we prove that this conjecture is true for loop spaces. Actually, we show that every quasi finite loop space is equivalent to a stably parallelizable…

This work introduces a new class of symmetric matrix structures, called harmonic structures, which enable the generation of all possible directed transitions $(x_i, x_{i+1})$ over a set of $n$ symbols, without internal repetitions. Unlike…

We introduce a notion of equivalence on tilings which is formulated in terms of their local structure. We compare it with the known concept of locally deriving one tiling from another and show that two tilings of finite type are…

Differential completions and compactifications of differential spaces are introduced and investigated. The existence of the maximal differential completion and the maximal differential compactification is proved. A sufficient condition for…

Closure spaces are a generalisation of topological spaces obtained by removing the idempotence requirement on the closure operator. We adapt the standard notion of bisimilarity for topological models, namely Topo-bisimilarity, to closure…

We consider a class of homogeneous self-similar sets with complete overlaps and give a sufficient condition for the Lipschitz equivalence between members in this class.

We review the construction and context of a stable homotopy refinement of Khovanov homology.

We introduce a model structure on the category of graphs, which is Quillen equivalent to the category of $\mathbb{Z}_2$-spaces. A weak equivalence is a graph homomorphism which induces a $\mathbb{Z}_2$-homotopy equivalence between their box…

We consider a fixed basis of a finitely generated free chain complex as a finite topological space and we present a sufficient condition for the singular homology of this space to be isomorphic with the homology of the chain complex.

Vietoris-Rips and degree Rips complexes are represented as homotopy types by their underlying posets of simplices, and basic homotopy stability theorems are recast in these terms. These homotopy types are viewed as systems (or functors),…

We present an intrinsic and concrete development of the subdivision of small categories, give some simple examples and derive its fundamental properties. As an application, we deduce an alternative way to compare the homotopy categories of…

In this paper, an algorithm building the effective homology version of the pushout of the simplicial morphisms $f:X\rightarrow Y$ and $g:X\rightarrow Z$, where $X,Y$ and $Z$ are simplicial sets with effective homology is presented.