Commuting involutions whose fixed point set consists of two special components
Volume 201 / 2008
Abstract
Let $F^n$ be a connected, smooth and closed $n$-dimensional manifold. We call $F^n$ a manifold with property $\mathcal{H}$ when it has the following property: if $N^m$ is any smooth closed $m$-dimensional manifold with $m>n$ and $T:N^m \to N^m$ is a smooth involution whose fixed point set is $F^n$, then $m=2n$. Examples of manifolds with this property are: the real, complex and quaternionic even-dimensional projective spaces $RP^{2n}$, $CP^{2n}$ and $HP^{2n}$, and the connected sum of $RP^{2n}$ and any number of copies of $S^n \times S^n$, where $S^n$ is the $n$-sphere and $n$ is not a power of $2$. In this paper we describe the equivariant cobordism classification of smooth actions $(M^m; {\mit\Phi})$ of the group $Z_2^k$ on closed smooth $m$-dimensional manifolds $M^m$ for which the fixed point set of the action consists of two components $K$ and $L$ with property $\mathcal{H}$, and where ${\rm dim}(K) < {\rm dim}(L)$. The description is given in terms of the set of equivariant cobordism classes of involutions fixing $K \cup L$.
