Involutions Fixing Two Copies of Projective Spaces Under Different Rings

Abstract

Consider the real, complex and quaternionic n-dimensional projective spaces, $\mathbb {R}P^n$, $\mathbb {C}P^n$ and $\mathbb {H}P^n$; to unify notation, write $K_dP^n$ for the real ($d=1$), complex ($d=2$) and quaternionic ($d=4$) n-dimensional projective space. Consider a pair (M, T), where M is a closed smooth manifold and T is a smooth involution defined on M. Write F for the fixed point set of T. In this paper we obtain the equivariant cobordism classification of the pairs (M, T) when F is of the form $F=K_dP^m \cup K_eP^n$, with $d<e$, in the cases $(m,n)=(odd,odd)$ and $(m,n)=(odd,even)$. As will be seen in the history described in Sect. 1, the case F = a disjoint union of projective spaces has an intense and still unfinished history in the literature; in particular, there are several results in the literature in which F is the union of two projective spaces, but with $d=e$; our paper is the first to consider unions of projective spaces under different rings. For example, if $d=e$, in the case $(m,n)=(odd,odd)$ all involution bounds, while for different rings there are nonbounding involutions.