Limiting cases of Boardman's five halves theorem

Abstract

AbstractThe famous five halves theorem of Boardman states that, if T: Mm → Mm is a smooth involution defined on a non-bounding closed smooth m-dimensional manifold Mm (m > 1) and ifis the fixed-point set of T, where Fj denotes the union of those components of F having dimension j, then 2m ≤ 5n. If the dimension m is written as m = 5k − c, where k ≥ 1 and 0 ≤ c < 5, the theorem states that the dimension n of the fixed submanifold is at least β(m), where β(m) = 2k if c = 0, 1, 2 and β(m) = 2k − 1 if c = 3, 4. In this paper, we give, for each m > 1, the equivariant cobordism classification of involutions (Mm, T), for which the fixed submanifold F attains the minimal dimension β(m).