On the homomorphism between the equivariant SK ring and the Burnside ring for involution (Q1074911)

Let G be a finite Abelian group. If M and N are smooth G-manifolds, then say \(M\sim N\) if the H-fixed point sets \(M^ H\) and \(N^ H\) have the same Euler characteristics for all \(H\leq G\). The Burnside ring A(G) of G is realized as equivalence classes [M] with sum corresponding to disjoint union and product to Cartesian product. Let \(SK_*^ G\) denote the G- equivariant ''cutting and pasting'' ring formed from smooth G-manifolds M; there is a natural ring homomorphism \(\phi\) : SK\({}^ G_*\to A(G)\). Set \(SK_*=SK_*^{(1)}\); then \(SK^ G_*\) is an \(SK_*\)-algebra. In this paper, the \(SK_*\)-algebra structure of \(SK_*^{Z_ 2}\) is determined explicitly as well as the kernel of the homomorphism \(\phi\) into \(A(Z_ 2)\).