Stable almost complex structures on certain $10$-manifolds

Abstract

Let $M$ be a $10$-dimensional closed oriented smooth manifold. Set $${\mathcal{D}}_M:=\{x\in H^2(M;\mathbb{Z}/2)\mid x^2+w_2(M)x\in {\rho }_2(T{H}^4(M;\mathbb{Z}))\}.$$ Suppose that $H_1(M;\mathbb{Z})=0$ and ${\mathcal{D}}_M\subset {\rho }_2(H^2(M;\mathbb{Z}))$. 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 $M$.

11 pages