Equivariant surgery with middle dimensional singular sets. II: Equivariant framed cobordism invariance

Abstract:

Let $G$ be a finite group and let $f : X \to Y$ be a degree 1, $G$-framed map such that $X$ and $Y$ are simply connected, closed, oriented, smooth manifolds of dimension $n = 2k \geqq 6$ and such that the dimension of the singular set of the $G$-space $X$ is at most $k$. In the previous article, assuming $f$ is $k$-connected, we defined the $G$-equivariant surgery obstruction $\sigma (f)$ in a certain abelian group. There it was shown that if $\sigma (f) = 0$ then $f$ is $G$-framed cobordant to a homotopy equivalence $f’ : X’ \to Y$. In the present article, we prove that the obstruction $\sigma (f)$ is a $G$-framed cobordism invariant. Consequently, the $G$-surgery obstruction $\sigma (f)$ is uniquely associated to $f : X \to Y$ above even if it is not $k$-connected.