The fixed-point property of $(2m-1)$-connected $4m$-manifolds

Abstract

Suppose that M is a $(2m - 1)$-connected smooth and compact manifold of dimension 4m. Assume that its intersection pairing is positive definite, and denote its signature by $\sigma$. Two notions are introduced. The first is that of a $(\xi ,\lambda )$-map $f:M \to M$ where $\xi \in K(M)$ and $\lambda$ an integer. It describes the concept of f preserving $\xi$ up to multiplication by $\lambda$ outside a point. The second notion is that of $\xi$ being sufficiently asymmetric. It describes in terms of the Chern class of $\xi$ the concept that the restrictions of $\xi$ to the 2m-spheres realizing a basis for ${H_{2m}}(M;Z)$ are sufficiently different so that no map which preserves $\xi$ can move the spheres among themselves. One proves that $(\xi ,\lambda )$-maps with $\xi$ being sufficiently asymmetric have fixed points, except possibly when $\sigma = 2$. On taking $\xi$ to be the complexification of the tangent bundle of M, one sees that mainfolds with sufficiently asymmetric tangent structures have the fixed point property with respect to a family of maps which includes diffeomorphisms. The question of the existence of $(\xi ,\lambda )$-maps as well as the question of the preservation of the fixed-point property under products are also discussed.