Stable almost complex structures on certain $10$-manifolds

TL;DR

This paper characterizes when certain 10-dimensional closed oriented smooth manifolds admit stable almost complex structures, based on their cohomology, characteristic classes, and specific algebraic conditions.

Contribution

It provides necessary and sufficient conditions for stable almost complex structures on 10-manifolds with particular cohomological properties, extending understanding of complex structures in high dimensions.

Findings

Criteria involving characteristic classes and cohomology for stable almost complex structures.

Conditions expressed in terms of the set _M and the second Stiefel-Whitney class.

Results applicable to 10-manifolds with trivial first homology and specific cohomological constraints.

Abstract

Let $M$ be a $10$-dimensional closed oriented smooth manifold. Set $$\mathcal{D}_{M} := \{ x \in H^{2}(M; \Z/2) \mid x^{2} + w_{2}(M) x \in \rho_{2} ( TH^{4}(M;\Z) ) \}.$$ Suppose that $H_{1}(M;\Z)=0$ and $\mathcal{D}_{M} \subset \rho_{2}( H^{2}(M; \Z) )$. Then the necessary and sufficient conditions for $M$ to admit a stable almost complex structure are determined in terms of the characteristic classes and cohomology ring of $M$.