Involutions whose fixed set has three or four components: a small codimension phenomenon (Q2890397)

In 1967, \textit{J. M. Boardman} proved the famous Five Halves Theorem, [Bull. Am. Math. Soc. 73, 136--138 (1967; Zbl 0153.25403)] which states: if \(M^m\) is a smooth closed \(n\)-dimensional manifold, and if \(T:M^m \to M^m\) is a smooth involution on \(M^m\) for which the fixed point set \(F = \displaystyle {\cup_{j =0}^{n}} F^j \) does not bound, then \(m \leq \displaystyle \frac{5}{2}n\) (here, \(F^j\) denotes the union of those components of \(F\) having dimension \(j\)). Inspired by the work of \textit{C. Kosniowski} and \textit{R. E. Stong} [Topology 17, 309--330 (1978; Zbl 0402.57005)] and \textit{D. C. Royster} [Indiana Univ. Math. J. 29, 267--276 (1980; Zbl 0406.57027)], \textit{P. L. Q. Pergher} introduced a class of problems where \(F = F^n \cup \{\text{point}\}\) improving Royster's result and in a joint work with \textit{R. E. Stong} [Transform. Groups 6, No. 1, 79--86 (2001; Zbl 0985.57017)], writing \(n=2^pq, \, p \geq 0\) and \(q\) odd, they showed that if \(p \leq q\) then \(k \leq n+p-q+1\) and if \(p>q\), then \(k \leq n+2^{p-q}\). In recent years, the third author of the present paper obtained several relevant results improving the Boardman bounds, by considering other additional conditions on \(F\), particularly where the fixed point set has the form \(F^n \cup F^j\), where \(F^j\) is indecomposable and \(n>j\).NEWLINENEWLINEIn this paper, the authors deal with cases involving small codimension phenomena, which means that the codimension of the \(n\)-dimensional component (dimension top) is not limited as a function of \(n\) and comparing with the previous articles of the third author, it is observed that it is the first time one deals with more than two components. Consider \((M,T)\) an involution with fixed point set having one of the forms \(F = F^n \cup F^3 \cup F^2 \cup \{\text{point}\}; F = F^n\cup F^3\cup F^2; F = F^n\cup F^3\cup \{\text{point}\}\) or \(F=F^n \cup F^3\), where \(n \geq 4\) even and suppose that the normal bundle to each \(F^j\) does not bound. Denoting by \(k\) the codimension of \(F^n\), the authors prove that \(k \leq 4\) and also that this bound is best possible, with the restriction \(n = 4t, \,\;t \geq 1\), for the first and third case.