Connected sums of almost complex manifolds

Abstract

In this paper, firstly, for some $4n$-dimensional almost complex manifolds $M_i,1\le i\le \alpha$, we prove that $\left({♯}_{i=1}^{\alpha }{M}_i\right)♯(\alpha -1)\mathbb{C}{P}^{2n}$ must admits an almost complex structure, where $\alpha$ is a positive integer. Secondly, for a $2n$-dimensional almost complex manifold $M$, we get that $M♯\overline{\mathbb{C}{P}^n}$ also admits an almost complex structure. At last, as an application, we obtain that $\alpha \mathbb{C}{P}^{2n}♯\beta \overline{\mathbb{C}{P}^{2n}}$ admits an almost complex structure if and only if $\alpha$ is odd.