good covers and simplicial maps - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T05:33:57Z http://mathoverflow.net/feeds/question/91663 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91663/good-covers-and-simplicial-maps good covers and simplicial maps Ulrich Pennig 2012-03-19T20:31:49Z 2012-03-19T20:31:49Z <p>Let $X$ be a paracompact topological space and choose a good cover $U_i$ of $X$. Remember that a good cover is one that consists of open subsets, such that each set $U_i$ is contractible and all finite intersections $U_{i_1} \cap U_{i_2} \cap \dots \cap U_{i_k}$ are either empty or contractible. </p> <p>This induces a simplicial topological space $U_{\bullet}$. Let $Z_{\bullet}$ be another simplicial topological space. I keep reading that any continuous map $$f \colon |U_{\bullet}| \to |Z_{\bullet}|$$ is homotopic to a simplicial one (e.g. in the sketched proof of theorem 4.5 <a href="http://arxiv.org/abs/math/0306027" rel="nofollow">here</a>). Is there any reference for this? How do I prove this? </p>