Topology of Almost Complex Structures on Six-Manifolds

TL;DR

This paper investigates the topology of the space of almost complex structures on six-manifolds, providing a new description via quotient spaces and computing their rational homotopy models and intersection numbers.

Contribution

It introduces a novel quotient space description of almost complex structures on six-manifolds and computes their rational homotopy models and intersection formulas.

Findings

Computed the rational homotopy minimal models of certain almost complex structures.

Derived a formula for the homological intersection number in terms of Chern classes.

Provided a new topological perspective on the space of almost complex structures.

Abstract

We study the space of (orthogonal) almost complex structures on closed six-dimensional manifolds as the space of sections of the twistor space for a given metric. For a connected six-manifold with vanishing first Betti number, we express the space of almost complex structures as a quotient of the space of sections of a seven-sphere bundle over the manifold by a circle action, and then use this description to compute the rational homotopy theoretic minimal model of the components that satisfy a certain Chern number condition. We further obtain a formula for the homological intersection number of two sections of the twistor space in terms of the Chern classes of the corresponding almost complex structures.