The fixed-point construction in equivariant bordism

Abstract:

Consider the bordism ${\Omega _ {\ast }}(G)$ of smooth G-actions. If K is a subgroup of G, with normalizer NK, there is a standard $NK/K$-action on ${\Omega _ {\ast }}(K)$(All, Proper). If M has a smooth G-action, a tubular neighborhood of the fixed set of K in M represents an element of ${\Omega _ {\ast }}(K){({\text {All, Proper}})^{NK/K}}$. One thus obtains the “fixed point homomorphism” $\phi$ carrying ${\Omega _ {\ast }}(G)$ to the sum of the ${\Omega _ {\ast }}(K){({\text {All, Proper}})^{NK/K}}$, summed over conjugacy classes of subgroups K. Let P be the collection of primes not dividing the order of G. We show that the P-localization of $\phi$ is an isomorphism, and give several applications.