An improvement of the five halves theorem of J. Boardman

Abstract

Let (Mm, T) be a smooth involution on a closed smooth m-dimensional manifold and F = ∪j=0nFj (n ≤ m) its fixed point set, where Fj 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 ≤ 5/2n. In this paper we obtain an improvement of the Five Halves Theorem when the top dimensional component of F, Fn, is nonbounding. Specifically, let ω = (i1, i2, …, ir) be a non-dyadic partition of n and sω(x1, x2, …, xn) the smallest symmetric polynomial over Z2 on degree one variables x1, x2, …, xn containing the monomial \(x_1^{i_1 } x_2^{i_2 } \cdots x_r^{i_r }\). Write sω(Fn) ∈ Hn(Fn, Z2) for the usual cohomology class corresponding to sω(x1, x2, …, xn), and denote by l(Fn) the minimum length of a nondyadic partition ω with sω(Fn) ≠ 0 (here, the length of ω = (i1, i2, …, ir) is r). We will prove that, if (Mm, T) is an involution for which the top dimensional component of the fixed point set, Fn, is nonbounding, then m ≤ 2n + l(Fn); roughly speaking, the bound for m depends on the degree of decomposability of the top dimensional component of the fixed point set. Further, we will give examples to show that this bound is best possible.