Combinatorial homotopy. II

Whitehead, J. H. C.

doi:10.1090/s0002-9904-1949-09213-3

Cited by 406 publications

(343 citation statements)

References 13 publications

Supporting

Mentioning

333

Contrasting

Unclassified

Order By: Relevance

“…Two homotopy equivalent spaces are weakly homotopy equivalent (the converse is not true in general but the Whitehead's theorem [39,40]) states that it is true in complexes (see Section 4.1).…”

Section: Algebraic Topologymentioning

confidence: 99%

Paths, Homotopy and Reduction in Digital Images

et al. 2010

View full text Add to dashboard Cite

The development of digital imaging (and its subsequent applications) has led to consider and investigate topological notions, well-defined in continuous spaces, but not necessarily in discrete/digital ones. In this article, we focus on the classical notion of path. We establish in particular that the standard definition of path in algebraic topology is coherent w.r.t. the ones (often empirically) used in digital imaging. From this statement, we retrieve, and actually extend, an important result related to homotopy-type preservation, namely the equivalence between the fundamental group of a digital space and the group induced by digital paths. Based on this sound definition of paths, we also (re)explore various (and sometimes equivalent) ways to reduce a digital image in a homotopy-type preserving fashion.

show abstract

Section: Algebraic Topologymentioning

confidence: 99%

Paths, Homotopy and Reduction in Digital Images

et al. 2010

View full text Add to dashboard Cite

show abstract

“…It is well known that strict categorical groups are the same as the crossed modules of Whitehead [41]. Moreover, braided strict categorical groups correspond to reduced 2-crossed modules and symmetric strict categorical groups to stable crossed modules (see [14,3]).…”

Section: Preliminariesmentioning

confidence: 99%

Simplicial Cohomology with Coefficients in Symmetric Categorical Groups

Carrasco

Martínez-Moreno

2004

Applied Categorical Structures

View full text Add to dashboard Cite

Abstract. In this paper we introduce and study a cohomology theory {H n (−, A)} for simplicial sets with coefficients in symmetric categorical groups A. We associate to a symmetric categorical group A a sequence of simplicial sets {K(A, n)} n≥0 , which allows us to give a representation theorem for our cohomology. Moreover, we prove that for any n ≥ 3, the functor K(−, n) is right adjoint to the functor ℘ n , where ℘ n (X • ) is defined as the fundamental groupoid of the n-loop complex n (X • ). Using this adjunction, we give another proof of how symmetric categorical groups model all homotopy types of spaces Y with π i (Y ) = 0 for all i = n, n + 1 and n ≥ 3; and also we obtain a classification theorem for those spaces:Mathematics Subject Classifications (2000): 18D10, 18G50, 18G60.

show abstract

“…. , Whitehead's ntypes [30] based on the concept of n-homotopy introduced in [17]. Algebraic models for n-types (the so called cat n -groups) were found in [24].…”

Section: Theorem Bmentioning

confidence: 99%

Topological Model Categories Generated by Finite Complexes

Chigogidze

Karasev

2003

Monatshefte f�r Mathematik

View full text Add to dashboard Cite

Abstract. Our main result states that for each finite complex L the category TOP of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely maps which induce isomorphisms of all [L]-homotopy groups. The concept of [L]-homotopy has earlier been introduced by the first author and is based on Dranishnikov's notion of extension dimension. As a corollary we obtain an algebraic characterization of [L]-homotopy equivalences between [L]-complexes. This result extends two classical theorems of J. H. C. Whitehead. One of them -describing homotopy equivalences between CW-complexes as maps inducing isomorphisms of all homotopy groups -is obtained by letting L = {point}. The other -describing n-homotopy equivalences between at most (n + 1)-dimensional CW-complexes as maps inducing isomorphisms of k-dimensional homotopy groups with k ≤ n -by letting L = S n+1 , n ≥ 0.

show abstract

Combinatorial homotopy. II

Cited by 406 publications

References 13 publications

Paths, Homotopy and Reduction in Digital Images

Paths, Homotopy and Reduction in Digital Images

Simplicial Cohomology with Coefficients in Symmetric Categorical Groups

Topological Model Categories Generated by Finite Complexes

Contact Info

Product

Resources

About