Almost complex manifolds with total Betti number three

TL;DR

This paper proves that for high-dimensional closed almost complex manifolds, the minimal total Betti number is four, confirming a conjecture except in dimension six, and characterizes simply connected complex manifolds with total Betti number three.

Contribution

It establishes the minimal total Betti number for almost complex manifolds in dimensions eight and higher and characterizes the unique simply connected complex manifold with total Betti number three.

Findings

Minimal total Betti number is four for dimensions ≥8.

The only simply connected closed complex manifold with total Betti number three is the complex projective plane.

Confirmed a conjecture of Sullivan in most dimensions.

Abstract

We show the minimal total Betti number of a closed almost complex manifold of dimension $2n\ge 8$ is four, thus confirming a conjecture of Sullivan except for dimension $6$. Along the way, we prove the only simply connected closed complex manifold having total Betti number three is the complex projective plane.