Improvements of the Five Halves Theorem of J. Boardman with respect to the decomposability degree

Abstract

Let $(M^m, T)$ be a smooth involution on a closed smooth $m$-dimensional manifold and $F = {\bigcup}^n_{j=0} F^j (n \lt m)$ its fixed point set, where $F^j$ denotes the union of those components of $F$ having dimension $j$. The famous Five Halves Theorem of J. Boardman, announced in 1967, establishes that, if $F$ is nonbounding, then $m \leq \frac{5}{2} n$; further, this estimative is best possible. In this paper we obtain improvements of this theorem, taking into account certain natural numbers which we call the decomposability degrees $\ell(F^j)$ of the nonbounding components $F^j$ of $F$ (see the definition in Section 1). Also, these improvements are obtained under assumptions on the set of dimensions occurring in $F$, which we denote $\pi_0(F)$. The main result of this paper is: suppose the involution $(M^m, T)$ has $\pi_0(F) = \{ 0, 1, \dots, j, n \}$, where $2 \leq j \lt n \lt m$ and $F^j$ is nonbounding. Write $\mathcal{M}(n - j)$ for the function of $n - j$ defined in the following way: writing $n - j = 2^p q$, where $q \geq 1$ is odd and $p \geq 0$, $M(n - j) = 2n + p - q + 1$ if $p \leq q$ and $M(n - j) = 2n + 2^{p-q}$ if $p \geq q$. Then $m \leq \mathcal{M}(n - j) + 2j + \ell (F^j)$. In addition, we develop a method to construct involutions $(M^m, T)$ with $\pi_0(F)$ as above, in some special situations, which in some cases will show that the above bound is best possible. This will provide some improvements of the general Five Halves Theorem $(\pi_0(F) = \{ i / 0 \leq i \leq n \} )$, by considering the particular case $j = n - 1$.