Nonexistence of Almost Complex Structures on the product $S^{2m}\times M$

Abstract

In this note we give a necessary condition for having an almost complex structure on the product $S^{2m}\times M$, where $M$ is a connected orientable closed manifold. We show that if the Euler characteristic $\chi (M)\ne 0$, then except for finitely many values of $m$, we do not have almost complex structure on $S^{2m}\times M$. In the particular case when $M=\mathbb{C}{\mathbb{P}}^n,n\ne 1$, we show that if $n\not\equiv 3(\mathrm{mod}4)$ then $S^{2m}\times \mathbb{C}{\mathbb{P}}^n$ has an almost complex structure if and only if $m=1,3$. As an application we obtain conditions on the nonexistence of almost complex structure on Dold manifolds.

arXiv comment9 pages

Cited by in corpus (1)

• On generalized Dold manifolds