积$S^{2m} \times M$上殆复结构的不存在性
Nonexistence of Almost Complex Structures on the product $S^{2m} \times M$
摘要：
本文给出了积$S^{2m} \times M$具有几乎复结构的一个必要条件，其中$M$是连通可定向闭流形.我们证明了如果欧拉特征$χ(M) \neq 0$，那么除了$m$的100个值之外，我们在$S^{2m} \times M$上没有几乎复的结构.在特定的情况下，当$M = \mathbb{C}\mathbb P^n, n \neq 1$，我们表明，如果$n \not \equiv 3 \pmod 4$然后$S^{2m} \times \mathbb C \mathbb P^{n}$有一个几乎复杂的结构当且仅当$m = 1,3$。作为应用，我们得到了Dold流形上几乎复结构不存在的条件。
英文摘要：
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 $χ(M) \neq 0$, 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.
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