Does a pointed homotopy equivalence between pointed $G$-spaces which is $G$-equivariant induce a (weak) homotopy equivalence on pointed Borel constructions? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T20:06:02Z http://mathoverflow.net/feeds/question/105161 http://www.creativecommons.org/licenses/by-nc/2.5/rdf 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/105161/does-a-pointed-homotopy-equivalence-between-pointed-g-spaces-which-is-g-equiv/105168#105168 Answer by Peter May for Does a pointed homotopy equivalence between pointed $G$-spaces which is $G$-equivariant induce a (weak) homotopy equivalence on pointed Borel constructions? Peter May 2012-08-21T15:44:05Z 2012-08-21T15:44:05Z <p>In the pointed Borel construction, you clearly mean $\wedge$ and not $\times$. Thus $$EG_+\wedge_G X = EG\times_G X/EG\times_G\ast.$$ Out of laziness, I'll assume that your $X$ and $Y$ are of the $G$-homotopy types of $G$-CW complexes. The based $G$-map $id\times f\colon EG\times X\longrightarrow EG\times Y$ is a homotopy equivalence on passage to $H$-fixed points for all $H\subset G$: the condition is empty unless $H=e$, when it is your hypothesis. Therefore $id\times f$ is a $G$-homotopy equivalence. Via the inclusions of $EG$ in source and target given by the basepoints of $X$ and $Y$, $id\times f$ is a map over $EG$ and therefore a $G$-homotopy equivalence over $EG$ since the inclusions of $EG$ in source and target are $G$-cofibrations by your well-pointed hypothesis. On passage to orbits over $G$ and quotient spaces, it follows that $$id\wedge_G f\colon EG_+\wedge_G X \longrightarrow EG_+\wedge_G Y$$ is a based homotopy equivalence.</p>