Involutions fixing

Let M m be a closed and smooth manifold with an involution having fixed point set of the form F n ∪ F 2 , where F n and F 2 are submanifolds with dimensions n and 2, respectively, and where 2 < n < m and F n ∪ F 2 does not bound. The main result of this paper is to establish the upper bound for m, for each n. The existence of these bounds is guaranteed by the famous 5 2 -theorem of J. Boardman, which establishes that, under the above hypotheses, m ⩽ 5 2 n .