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

TL;DR

This paper establishes necessary conditions for the existence of almost complex structures on products of even-dimensional spheres and manifolds, showing nonexistence results based on Euler characteristic and specific manifold cases.

Contribution

It provides new criteria for when almost complex structures cannot exist on products of spheres with certain manifolds, including explicit conditions for complex projective spaces.

Findings

Almost complex structures are rare on products $S^{2m} imes M$ when $ ext{chi}(M) eq 0$.

Existence of almost complex structures on $S^{2m} imes ext{CP}^n$ depends on specific values of $m$ and $n$.

Conditions for nonexistence of almost complex structures on Dold manifolds are derived.

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 $\chi(M) \neq 0$, 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 $M = \mathbb{C}\mathbb P^n, n \neq 1$, we show that if $n \not \equiv 3 \pmod 4$ 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.