Almost Complex Structures on (n-1)-connected 2n-manifolds

TL;DR

This paper establishes necessary and sufficient conditions for (n-1)-connected 2n-manifolds to admit almost complex structures, using Wall's invariants to characterize the manifolds' properties.

Contribution

It provides a complete characterization of when such high-dimensional manifolds admit almost complex structures based on Wall's invariants.

Findings

Derived necessary and sufficient conditions for almost complex structures

Connected the existence of structures to Wall's invariants

Extended understanding of complex structures on high-dimensional manifolds

Abstract

Let M be a closed (n-1)-connected 2n-dimensional smooth manifold with n > 2. In terms of the system of invariants for such manifolds introduced by Wall, we obtain necessary and sufficient conditions for M to admit an almost complex structure.