General gluing theorem for adjunction spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T15:17:37Z http://mathoverflow.net/feeds/question/96071 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96071/general-gluing-theorem-for-adjunction-spaces General gluing theorem for adjunction spaces VCF 2012-05-05T15:57:59Z 2012-05-06T17:49:32Z <p>Consider the following interesting theorem:(7.5.7, p.294 in <em>Topology and Groupoids</em> by Ronald Brown)</p> <p><strong>Gluing theorem for adjunction spaces:</strong> <em>Suppose that we have the following commutative diagram of topological spaces and continuous maps:</em></p> <p><img src="http://i48.tinypic.com/sdixrc.jpg" alt="alt text"></p> <p><em>where</em> $\varphi_{A}$, $\varphi_{X}$ <em>and</em> $\varphi_{Y}$ <em>are homotopy equivalences, and the inclusions</em> $i$ <em>and</em> $i'$ <em>are closed cofibrations. Then the map</em></p> <p>$$\varphi:X\cup_{f}Y\rightarrow X'\cup_{f\phantom{l}'}Y'$$</p> <p><em>induced by</em> $\varphi_{A}$, $\varphi_{X}$ <em>and</em> $\varphi_{Y}$ <em>is a homotopy equivalence</em>. </p> <p><a href="http://mathoverflow.net/questions/94830" rel="nofollow">In this post</a> I asked a question that would become a corollary if the gluing theorem were true for arbitrary cofibrations $i$ and $i'$. <strong>I would like to know if this is the case. Otherwise, does anybody know of a counterexample?</strong></p> <p>Perhaps, it is possible that although the map $\varphi$ may not in general be a homotopy equivalence when the cofibrations $i$ and $i'$ are arbitrary, the adjunction spaces $Y\cup_{f}X$ and $Y'\cup_{f'}X'$ will nonetheless be homotopy equivalent. A possible approach to this would be to try to find a third space containing both as deformation retracts. </p> <p><strong>Note on notation:</strong> That an inclusion map $j:B\rightarrow Z$ is a closed cofibration means that it is a cofibration (i.e., the pair $(B,Z)$ has the homotopy extension property) and the set $B$ is closed in $X'$.</p> <p>Thank you</p> http://mathoverflow.net/questions/96071/general-gluing-theorem-for-adjunction-spaces/96116#96116 Answer by Karol Szumiło for General gluing theorem for adjunction spaces Karol Szumiło 2012-05-06T05:59:22Z 2012-05-06T05:59:22Z <p>Yes, this is true even for not necessarily closed cofibrations. If you want a single source that gives a complete proof, then the only one that comes to my mind is this <a href="http://arxiv.org/abs/math/0610009v4" rel="nofollow">preprint</a>.</p> <p>Definition 1.1.1 introduces cofibration categories and then Lemma 1.4.1 says that the desired result holds in any cofibration category. Section 3.1 contains a detailed proof that the category of topological spaces equipped with Hurewicz cofibrations and homotopy equivalences is a cofibration category and thus the lemma applies. The crux of the matter is that acyclic cofibrations are closed under pushouts and this follows from a classical result of Dold (Lemma 3.1.9) that acyclic cofibrations admit deformation retractions, which doesn't depend on closedness.</p>