Involutions fixing many components: a small codimension phenomenon (Q1684914)

Let \(T: M\to M\) be an involution on a smooth closed \(m\)-manifold \(M\). The fixed point set \(F\) of \(T\) is generally a disjoint union of many components with different dimensions. Let \(\pi_0(F)\) denote the set of dimensions occurring in \(F\). Assume that \(n\) is the maximal number in \(\pi_0(F)\), i.e., the dimension of the component of \(F\) of largest dimension. If \(M\) is not a boundary, then the famous \({5\over 2}\)-Theorem of Boardman tells us that \(m\leq {5\over 2}n\) or \(k\leq {3\over 2}n\), where \(k=m-n\). The paper under review considers the general case regardless of whether \(M\) is a boundary or not, and studies what the best possible upper bound of \(k\) is. The problem was first discussed by the second author in [Mat. Contemp. 13, 269--275 (1997; Zbl 0920.57009)]. Some works in special cases show that there exists the so called \textit{small codimension phenomenon}, which is quite interesting. Specifically speaking, the upper bound of \(k\) depends upon the existence of small dimensions in \(\pi_0(F)\). For example, one can see from \textit{E. M. Barbaresco} and the present authors [Math. Scand. 110, No. 2, 223--234 (2012; Zbl 1256.57025)] that if \(\pi_0(F)=\{3, n\}, \{0,3,n\}, \{2,3,n\}, \{0,2,3,n\}\) with \(n\geq 4\), then \(k\leq 4\). In particular, there are involutions showing that this bound is best possible in the cases \(\pi_0(F)=\{3, n\}, \{2,3,n\}\), and in the cases \(\pi_0(F)=\{0,3,n\}, \{0,2,3,n\}\) with \(n\) of the form \(n = 4t, t \geq 1\). The paper under review characterizes small codimension phenomena in the following cases: (i) \(0 \in \pi_0(F)\), and all the other components of \(F\) (including the top-dimensional, with dimension \(n\)) are odd-dimensional; (ii) \(1\in\pi_0(F)\), and all the other components of \(F\) (including the top-dimensional) are even-dimensional. Then \(k \leq 1\) in the first case, and \(k \leq 2\) in the second case. Based upon this, the authors of paper under review pose the conjecture: if either there is an odd number \(0<j <n\), so that \(j \in \pi_0(F)\), and all the other components of \(F\) (including the top-dimensional) are even-dimensional or there is an even number \(0 \leq j < n\), so that \(j\in \pi_0(F)\), and all the other components of \(F\) (including the top-dimensional) are odd-dimensional, then, \(k \leq j +1\), and the result is best possible. In addition, the authors of paper under review show by some examples that if this conjecture is valid, it is best possible.