User richard manthorpe - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T18:06:27Z http://mathoverflow.net/feeds/user/25828 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106022/when-is-the-inclusion-of-a-relative-mapping-space-into-a-mapping-space-a-cofibrat When is the inclusion of a relative mapping space into a mapping space a cofibration? Richard Manthorpe 2012-08-31T10:02:51Z 2012-08-31T19:51:15Z <p>Let $(X,A)$ and $(Y,B)$ be pairs of spaces and subspaces, let $\operatorname{Map}(X,Y)$ the space of maps $f:X\to Y$ equipped with the compact-open topology and let $\operatorname{Map}(X,A;Y,B)$ be the subspace of maps $f:X\to Y$ such that $f(A)\subseteq B$. Suppose the inclusions $A\hookrightarrow X$ and $B\hookrightarrow Y$ are cofibrations, would that be enough to ensure the inclusion $\operatorname{Map}(X,A;Y,B)\hookrightarrow\operatorname{Map}(X,Y)$ is a cofibration or are other conditions needed?</p> <p>In particular if $X$ and $Y$ are well-pointed is the inclusion of the based mapping space $\operatorname{Map}_*(X,Y)\hookrightarrow\operatorname{Map}(X,Y)$ a cofibration?</p> <p>It seems like this should be true for reasonably nice spaces and there are similar results. I know, for example, that if $B\hookrightarrow Y$ is a closed cofibration and $X$ is compact Hausdorff then the inclusion $\operatorname{Map}(X,B)\hookrightarrow\operatorname{Map}(X,A;Y,B)$ is a cofibration. In particular this makes based mapping spaces with compact Hausdorff domain well-pointed.</p> http://mathoverflow.net/questions/105161/does-a-pointed-homotopy-equivalence-between-pointed-g-spaces-which-is-g-equiv Does a pointed homotopy equivalence between pointed $G$-spaces which is $G$-equivariant induce a (weak) homotopy equivalence on pointed Borel constructions? Richard Manthorpe 2012-08-21T14:21:34Z 2012-08-21T15:52:13Z <p>Let $G$ be a topological group and let $X$ and $Y$ be connected, well-pointed $G$-spaces. Suppose $f:X\to Y$ is a pointed homotopy equivalence and a $G$-equivariant map (but not an equivariant homotopy equivalence). I know that $f$ induces a (weak) homotopy equivalence on the Borel constructions, $EG\times_G X\to EG\times_G Y$, but what about the induced map on the <em>pointed</em> Borel constructions, $EG_+\wedge_G X\to EG_+\wedge_G Y$? Is it a homotopy equivalence too? As far as I can see it is a homology equivalence and a stable homotopy equivalence but I would like a stronger result.</p> http://mathoverflow.net/questions/106022/when-is-the-inclusion-of-a-relative-mapping-space-into-a-mapping-space-a-cofibrat/106038#106038 Comment by Richard Manthorpe Richard Manthorpe 2012-09-01T12:23:39Z 2012-09-01T12:23:39Z Thanks Mark, that's exactly the kind of argument I've been looking for! http://mathoverflow.net/questions/105161/does-a-pointed-homotopy-equivalence-between-pointed-g-spaces-which-is-g-equiv/105168#105168 Comment by Richard Manthorpe Richard Manthorpe 2012-08-21T17:41:50Z 2012-08-21T17:41:50Z Thank you that is exactly what I was looking for. And I guess the result for the scanning map is indeed for $G$-CW complexes. http://mathoverflow.net/questions/105161/does-a-pointed-homotopy-equivalence-between-pointed-g-spaces-which-is-g-equiv/105168#105168 Comment by Richard Manthorpe Richard Manthorpe 2012-08-21T16:04:34Z 2012-08-21T16:04:34Z Thanks, I did mean $\wedge$, I have updated the question. What if the spaces are not of the G-homotopy types of G-CW complexes? In particular I am thinking of configuration spaces and the scanning map - I have seen this claimed in a paper by B&#246;digheimer and Madsen when $G$ is a compact lie group.