Two Commuting Involutions Fixing F ⁿ ∪ F ^n-1

Abstract

Consider G=Z ²₂ as the group generated by two commuting involutions, and let $\Phi: G \times M \mapsto M$ be a smooth G-action on a smooth and closed manifold M. Suppose that the fixed point set of Φ consists of two connected components, F ⁿ and F ^n-1, with dimensions n and n−1, respectively. In this paper we prove that, if in the fixed data of Φ at least two eigenbundles over F ⁿ have dimension greater than n, and at least one eigenbundle over F ^n-1 has dimension greater than n−1, then the action (M,Φ) bounds equivariantly.It is well known that, if $T: M^m \mapsto M^m$ is a smooth involution on a smooth and closed m-dimensional manifold M ^m such that the fixed point set of T has constant dimension n, and if m > 2n, then (M ^m,T) bounds equivariantly; this fact was proved by R. E. Stong and C. Kosniowski 27 years ago. As a consequence of our result, we will see that the same fact is true when, besides n-dimensional components, the fixed point set contains additional (n−1)-dimensional components.