Some remarks on stable almost complex structures on manifolds

Abstract

Let $X$ be an $(8k+i)$-dimensional pathwise connected $CW$-complex with $i=1$ or $2$ and $k\ge0$, $\xi$ be a real vector bundle over $X$. Suppose that $\xi$ admits a stable complex structure over the $8k$-skeleton of $X$. Then we get that $\xi$ admits a stable complex structure over $X$ if the Steenrod square $$\mathrm{Sq}^{2}\colon H^{8k-1}(X;\mathbb{Z}/2)\rightarrow H^{8k+1}(X;\mathbb{Z}/2)$$ is surjective. As an application, let $M$ be a $10$-dimensional manifold with no $2$-torsion in $H_{i}(M;\mathbb{Z})$ for $i=1,2,3$, and no $3$-torsion in $H_{1}(M;\mathbb{Z})$. Suppose that the Steenrod square $$\mathrm{Sq}^{2}\colon H^{7}(M;\mathbb{Z}/2)\rightarrow H^{9}(M;\mathbb{Z}/2)$$ is surjective. Then the necessary and sufficient conditions for the existence of a stable almost complex structure on $M$ are given in terms of the cohomology ring and characteristic classes of $M$.