Dipaths and dihomotopies in a cubical complex

Abstract

In the geometric realization of a cubical complex without degeneracies, a $\Box$-set, dipaths and dihomotopies may not be combinatorial, i.e., not geometric realizations of combinatorial dipaths and equivalences. When we want to use geometric/topological tools to classify dipaths on the 1-skeleton, combinatorial dipaths, up to dihomotopy, and in particular up to combinatorial dihomotopy, we need that all dipahts are in fact dihomotopic to a combinatorial dipath. And moreover that two combinatorial dipaths which are dihomotopic are then combinatorially dihomotopic. We prove that any dipath from a vertex to a vertex is dihomotopic to a combinatorial dipath, in a non-selfintersecting $\Box$-set. And that two combinatorial dipaths which are dihomotopic threough a non-combinatorial dihomogopy are in fact combinatorially dihomotopic, in a geometric $\Box$-set. Moreover, we prove that in a geometric $\Box$-set, the d-homotopy introdced in [M. Grandis (2003)] coincides with the dihomotopy in [L. Fajstrup, E. Goubault, M. Raussen (1999)]. Udgivelsesdato: AUGIn the geometric realization of a cubical complex without degeneracies, a $\Box$-set, dipaths and dihomotopies may not be combinatorial, i.e., not geometric realizations of combinatorial dipaths and equivalences. When we want to use geometric/topological tools to classify dipaths on the 1-skeleton, combinatorial dipaths, up to dihomotopy, and in particular up to combinatorial dihomotopy, we need that all dipahts are in fact dihomotopic to a combinatorial dipath. And moreover that two combinatorial dipaths which are dihomotopic are then combinatorially dihomotopic. We prove that any dipath from a vertex to a vertex is dihomotopic to a combinatorial dipath, in a non-selfintersecting $\Box$-set. And that two combinatorial dipaths which are dihomotopic threough a non-combinatorial dihomogopy are in fact combinatorially dihomotopic, in a geometric $\Box$-set. Moreover, we prove that in a geometric $\Box$-set, the d-homotopy introdced in [M. Grandis (2003)] coincides with the dihomotopy in [L. Fajstrup, E. Goubault, M. Raussen (1999)]