Equivariant surgery with middle dimensional singular sets. II: Equivariant framed cobordism invariance (Q2706618)

The article is a continuation of the work of \textit{A. Bak} and the author [Forum Math. 8, No. 3, 267-302 (1996; Zbl 0860.57028)] where for a \(G\)-framed map \(f:X\to Y\) of degree 1, a \(G\)-equivariant surgery obstruction \(\sigma(f)\) is defined as an element of a certain abelian group, under the assumption that \(G\) is a finite group, \(X\) and \(Y\) are two closed, oriented, simply connected, smooth \(G\)-manifolds of dimension \(n=2k\geq 6\), the dimension of the singular set of \(X\) is at most \(k\), and the map \(f\) is \(k\)-connected. Bak and Morimoto show that if \(\sigma(f)=0\), then \(f\) is \(G\)-framed cobordant to a homotopy equivalence \(X'\to Y\). In the article under review, the author shows that even if \(f\) is not \(k\)-connected, one may define the obstruction \(\sigma(f)\). This follows from the main result of the article asserting that the obstruction \(\sigma(f)\) is a \(G\)-framed cobordism invariant. As the author notes, this result can be applied for constructing interesting smooth \(G\)-actions on manifolds such as disks and spheres.