${ Z_2^k}$-actions with a special fixed point set
Volume 186 / 2005
Abstract
Let $F^n$ be a connected, smooth and closed $n$-dimensional manifold satisfying the following property: if $N^m$ is any smooth and 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 $m=2n$. We describe the equivariant cobordism classification of smooth actions $(M^m; {\mit \Phi })$ of the group $G=Z_2^k$ on closed smooth $m$-dimensional manifolds $M^m$ for which the fixed point set of the action is a submanifold $F^n$ with the above property. This generalizes a result of F.~L. Capobianco, who obtained this classification for $F^n={\mathbb R}P^{2r}$ (P. E. Conner and E. E. Floyd had previously shown that ${\mathbb R}P^{2r}$ has the property in question). In addition, we establish some properties concerning these $F^n$ and give some new examples of these special manifolds.
