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 pointed 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.
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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.
answered Aug 21, 2012 at 15:44
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$\begingroup$ 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 Bödigheimer and Madsen when $G$ is a compact lie group. $\endgroup$ Commented Aug 21, 2012 at 16:04
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1$\begingroup$ $G$-CW isn't too restrictive (includes all smooth $G$-manifolds when $G$ is compact Lie). But here is an argument for weak equivalence without that assumption. A comparison of the Borel bundles for $X$ and $Y$ ($X\to EG\times_G X \to BG$) shows that the map of Borel constructions is a weak equivalence. The reduced Borel construction is the pushout of the maps $BG \to \ast$ and $BG \to EG\times_G X$. The latter map is a cofibration, so the gluing lemma for weak equivalences gives that the map of reduced Borel constructions is a weak equivalence. $\endgroup$ Commented Aug 21, 2012 at 17:24
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$\begingroup$ 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. $\endgroup$ Commented Aug 21, 2012 at 17:41