On the minimal sum of Betti numbers of an almost complex manifold

TL;DR

This paper characterizes the dimensions in which rational homology spheres can admit almost complex structures, showing they occur only in dimensions two and six, and explores the Betti number sums of such manifolds.

Contribution

It proves that only in dimensions two and six can rational homology spheres admit almost complex structures and classifies the Betti number sums of these manifolds.

Findings

Rational homology spheres admit almost complex structures only in dimensions two and six.

Infinite examples of six-dimensional rational homology spheres with almost complex structures are provided.

Manifolds with sum of Betti numbers three must have dimension a power of two.

Abstract

We show that the only rational homology spheres which can admit almost complex structures occur in dimensions two and six. Moreover, we provide infinitely many examples of six-dimensional rational homology spheres which admit almost complex structures, and infinitely many which do not. We then show that if a closed almost complex manifold has sum of Betti numbers three, then its dimension must be a power of two.