Question

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.

Answer 1

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.