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

TL;DR

This paper proves that the Yamabe invariant of certain 4k-manifolds remains unchanged when connected sums with quaternionic projective spaces or Cayley planes are performed, extending understanding of scalar curvature invariants under topological modifications.

Contribution

It establishes invariance of the Yamabe invariant under connected sums with quaternionic projective spaces and Cayley planes for specific manifolds.

Findings

Y(M)#l HP^k#m HP^k = Y(M) for nonpositive Y(M)

Y(M)#l CaP^2#m CaP^2 = Y(M) when k=4

Invariance holds for certain topological surgeries on 4k-manifolds

Abstract

Let $M$ be a smooth closed $4k$-manifold whose Yamabe invariant $Y(M)$ is nonpositive. We show that $$Y(M\sharp l \Bbb HP^k\sharp m \bar{\Bbb HP^k})=Y(M),$$ where $l,m$ are nonnegative integers, and $\Bbb HP^k$ is the quaternionic projective space. When $k=4$, we also have $$Y(M\sharp l CaP^2\sharp m \bar{CaP^2})=Y(M),$$ where $CaP^2$ is the Cayley plane.