Rational spheres and double disk bundles

Abstract

A manifold $M$ is said to be a double disk bundle if it can be decomposed as a union of two disk bundles glued together by a diffeomorphism of their boundaries. We show that if $M^n$ is a closed simply connected $n$-manifold with $n$ even which is simultaneously a double disk bundle and a rational homology sphere, then $M$ must be homeomorphic to a sphere. In addition, we show that in any dimension, if $M$ is a highly connected rational homology sphere which supports a double disk bundle structure, then its "middle" cohomlogy group must be cyclic.