Fixed sets of framed $G$-manifolds

Abstract:

This note describes restrictions on the framed bordism class of a framed manifold $Y$ in order that it be the fixed set of some framed $G$-manifold $M$ with $G$ a finite group. These results follow from a recently proved generalization of the Segal conjecture, and imply, in particular, that if $M$ is a framed $G$-manifold of sufficiently high dimension, and if $G$ is a $p$-group, then the number of “noncancelling” fixed points is either zero or approaches infinity as the dimension of $M$ goes to infinity. Conversely, we give sufficient conditions on the framed bordism class of a manifold $Y$ that it be the fixed set of some framed $G$-manifold $M$ of arbitrarily high dimension.