An improvement of the five halves theorem of J. Boardman (Q1760340)

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 \(F = \bigcup_{j =0}^{n} F^j\) does not bound, then \(m \leq \frac{5}{2}n\) (here, \(F^j\) denotes the union of those components of \(F\) having dimension \(j\)). In this paper, the author obtains improvements of this theorem, specifically for the case where the top dimension component \(F^n\) of \(F\) is nonbounding, and taking into account a cobordism invariant associated to \(F^n\), introduced by the author and called the \textit{decomposability degree of \(F^n\)}. This invariant is described as follows: Let \(s_\omega(x_1,x_2, \dots ,x_n)\) be the smallest symmetric polynomial over \(\mathbb{Z}_2\) on degree one variables \(x_1, x_2 , \dots , x_n\) containing the monomial \(x_{1}^{i_1}, x_{2}^{i_2}, \dots , x_{n}^{i_n}\), where \(\omega= (i_1, i_2, \dots , i_n)\) is a non-dyadic partition of \(n\). Consider \(s_\omega(F) \in H^n(F^n, \mathbb{Z}_2)\) the usual cohomology class corresponding to \(s_\omega(x_1,x_2, \dots ,x_n)\). Then the decomposability degree of \(F^n\) will be \(\ell(F^n),\) the minimum length of a non-dyadic partition \(\omega\) for which \(s_\omega(F)\) evaluated on the fundamental \(Z_2\)-homology class of \(Fn\) is nonzero. The following relevant result is proved by the author: if \(F^n\) is nonbounding, then \(m \leq 2n + \ell(F^n)\), and this estimate is best possible. For \(n\) even, this result gives the same bound of the Five Halves Theorem, but for \(n=2k+1\) odd, it is proved that \(m \leq 5k +1\), while the Five Halves Theorem gives that \(m \leq 5k +2\). Also, in the special case where \(n = 2k\) and \(\ell(F^n) <k\) and \(n=2k+1\) and \(\ell(F^n) <k-1\), the result of this paper provides effective improvements for the Five Halves Theorem.