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.



Does a pointed homotopy equivalence between pointed $G$-spaces which is $G$-equivariant induce a (weak) homotopy equivalence on pointed Borel constructions?
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.


