Involutions Fixing Fn ∪ ﹛Indecomposable﹜

Abstract

AbstractLet Mm be an m-dimensional, closed and smooth manifold, equipped with a smooth involution T : Mm → Mm whose fixed point set has the form Fn ∪ Fj, where Fn and Fj are submanifolds with dimensions n and j, Fj is indecomposable and n > j. Write n – j = 2pq, where q ≥ 1 is odd and p ≥ 0, and set m(n– j) = 2n + p–q +1 if p ≤ q +1 and m(n– j) = 2n +2p–q if p ≥ q. In this paper we show that m ≤ m(n – j) + 2j + 1. Further, we show that this bound is almost best possible, by exhibiting examples (Mm(n–j)+2j, T) where the fixed point set of T has the form Fn ∪ Fj described above, for every 2 ≤ j < n and j not of the form 2t – 1 (for j = 0 and 2, it has been previously shown that m(n – j) + 2 j is the best possible bound). The existence of these bounds is guaranteed by the famous 5/2-theorem of J. Boardman, which establishes that under the above hypotheses .