Dipaths and dihomotopies in a cubical complex
نویسنده
چکیده
In the geometric realization of a cubical complex without degeneracies, a 2-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 dipaths 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 2-set. And that two combinatorial dipaths which are dihomotopic through a non combinatorial dihomotopy are in fact combinatorially dihomotopic, in a geometric 2-set. Moreover, we prove that in a geometric 2-set, the d-homotopy introduced in [6] coincides with the dihomotopy in [5]
منابع مشابه
Dicovering Spaces
For a local po-space X and a base point x0 ∈ X, we define the universal dicovering space Π : X̃x0 → X. The image of Π is the future ↑ x0 of x0 in X and X̃x0 is a local po-space such that | π 1 (X̃, [x0], x1)| = 1 for the constant dipath [x0] ∈ Π−1(x0) and x1 ∈ X̃x0 . Moreover, dipaths and dihomotopies of dipaths (with a fixed starting point) in ↑ x0 lift uniquely to X̃x0 . The fibers Π −1(x) are dis...
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