Involutions fixing \(F^n \cup F^3\) (Q1705770)

The aim of this paper is to determine bounds on the dimension of manifolds with certain fixed-point sets under involutions. Let \(M^m\) be a smooth and closed \(m\)-dimensional manifold equipped with a smooth involution whose fixed-point set is \(F= \cup_{j=0}^n F^j\), where \(F^j\) denotes the union of those components of \(F\) having dimension \(j\). Up to equivariant cobordism, each \(F^j\) can be assumed to be connected and \(F\) can be assumed not to have a component whose normal bundle bounds. If \(n\) is the dimension of the component of \(F\) of maximal dimension, then \(m \leq 5n/2\) by the famous Five Halves Theorem of J. Boardman. In this paper, the authors consider the case when the fixed-point set \(F\) is a union of components of dimension \(n\) and \(3\). Let \(M^m\) be a closed smooth manifold equipped with a smooth involution having fixed point set \(F=F^n \cup F^3\), where \(F^n\) and \(F^3\) are submanifolds with dimensions \(n\) and \(3\), respectively, with \(3 < n < m\) and the normal bundles over \(F^n\) and \(F^3\) being nonbounding. In [\textit{E. M. Barbaresco} et al., Math. Scand. 110, No. 2, 223--234 (2012; Zbl 1256.57025)], it is proved that, when \(n\) is even, then \(m \leq n + 4\). The authors refer to this as a small codimension phenomenon. In this paper, the authors investigate this problem for the remaining case of \(n\) odd and show that in this case \(m \leq \mathcal{M}(n - 3) + 6\). Here \(\mathcal{M}(n)\) is the Stong-Pergher number given as follows: Let \(n = 2^p q\), where \(p \geq 0\) and \(q\) is odd. Then \(\mathcal{M}(n) = 2n + p - q + 1\) if \(p \leq q\) and \(\mathcal{M}(n) = 2n + 2^{p-q}\) if \(p > q\).