Connected sums with HP^n or CaP^2 and the Yamabe invariant

Abstract

Let $M$ be a smooth closed $4k$-manifold whose Yamabe invariant $Y(M)$ is nonpositive. We show that $$Y(M♯l\mathbb{H}{P}^k♯m\overline{\mathbb{H}{P}^k})=Y(M),$$ where $l,m$ are nonnegative integers, and $\mathbb{H}{P}^k$ is the quaternionic projective space. When $k=4$, we also have $$Y(M♯lCa{P}^2♯m\overline{Ca{P}^2})=Y(M),$$ where $Ca{P}^2$ is the Cayley plane.