Subject: colimits and homotopy From: "Claude Schochet" Date: Mon, 6 Sep 2004 21:25:38 -0400 For toplist: Everybody believes that homotopy commutes with direct limits. Does it in general? To be precise, suppose that X is the direct limit of a directed (not necessarily countable) family X_\alpha of topological spaces with structural maps not necessarily 1-1. Then one would like that the natural map dirlim \pi _*(X_\alpha ) \to \pi _*(X) would be an isomorphism. If I assume nothing about the spaces and nothing about the structural maps other than continuity, is this true? Obviously the key point is that any map from a compact K to X should factor through some X_\alpha. This is true, for instance, if we have a countable increasing union with X having the weak topology and each X_n closed in the next, by Whitehead 1.6.3. Is that the best result? Does it help to assume that each space is the homotopy type of a CW complex? In the case we're looking at we have X_\alpha = F(W_\alpha , U_n(C)) (function space of maps) where {W_\alpha } is an inverse system of finite CW - complexes (nerves of finite covers of W = invlim W_\alpha so we are really looking at dirlim \pi _*(F(W_\alpha , U_n(C)) ) \to \pi _*(X Thanks. Claude