Rational spheres and double disk bundles

TL;DR

This paper proves that certain highly connected rational homology spheres with double disk bundle structures are homeomorphic to spheres, and characterizes their middle cohomology groups.

Contribution

It establishes topological constraints on rational homology spheres that admit double disk bundle decompositions, extending understanding of their structure.

Findings

A simply connected even-dimensional rational homology sphere with a double disk bundle structure is homeomorphic to a sphere.

In any dimension, highly connected rational homology spheres with a double disk bundle have cyclic middle cohomology.

The results connect geometric decompositions with topological and homological properties of manifolds.

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.