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In this paper, the existence of almost complex structures on connected sums of almost complex manifolds and complex projective spaces are investigated. Firstly, we show that if M is a 2n-dimensional almost complex manifold, then so is \(M\sharp \overline{{\mathbb {C}}P^{n}}\), where \(\overline{{\mathbb {C}}P^{n}}\) is the n-dimensional complex projective space with the reversed orientation. Secondly, for any positive integer \(\alpha \) and any 4n-dimensional almost complex manifolds \(M_{i}, ~1\le i \le \alpha \), we prove that \(\left( \sharp _{i=1}^{\alpha } M_{i}\right) \sharp (\alpha {-}1) {\mathbb {C}}P^{2n}\) must admit an almost complex structure. At last, as an application, we obtain that \(\alpha {\mathbb {C}}P^{2n}\sharp ~\beta \overline{{\mathbb {C}}P^{2n}}\) admits an almost complex structure if and only if \(\alpha \) is odd.

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We record an answer to the question “In which dimensions is the connected sum of two closed almost complex manifolds necessarily an almost complex manifold?”. In the process of doing so, we are naturally led to ask “For which values of ℓ is the connected sum of ℓ closed almost complex manifolds necessarily an almost complex manifold?”. We answer this question, along with its non-compact analogue, using obstruction theory and Yang's results on the existence of almost complex structures on (n−1)-connected 2n-manifolds. Finally, we partially extend Datta and Subramanian's result on the nonexistence of almost complex structures on products of two even spheres to rational homology spheres by using the index of the twisted spinc Dirac operator.

We show that the m-fold connected sum $m\#\mathbb{C}\mathbb{P}^{2n}$ admits an almost complex structure if and only if m is odd.

Dedicated to Professor Shiing-shen Chern on his 91st birthday To generalize the Hopf index theorem and the Atiyah–Dupont vector fields theory, one is interested in the following problem: for a real vector bundle E over a closed manifold M with rank E = dim M , whether there exist two linearly independent cross sections of E? We provide, among others, a complete answer to this problem when both E and M are orientable. It extends the corresponding results for E = T M of Thomas, Atiyah, and Atiyah–Dupont. Moreover we prove a vanishing result of a certain mod 2 index when the bundle E admits a complex structure. This vanishing result implies many known famous results as consequences. Ideas and methods from obstruction theory, K-theory and index theory are used in getting our results.

Let M be a closed (n-1)-connected 2n-dimensional smooth manifold with n > 2.
In terms of the system of invariants for such manifolds introduced by Wall, we
obtain necessary and sufficient conditions for M to admit an almost complex
structure.

Let M be the blow-up of a complex manifold along a submanifold. We determine
the integral cohomology ring and obtain a formula for the Chern classes of M.
As applications we determine the cohomology rings for the varieties of complete
conics and complete quadrices in 3-space, and justify two enumerative results
due to Schubert [S1].

In this paper we prove the following theorem: S2p×S2q allows an almost complex structure if and only if (p, q) = (1,1), (1,2), (2,1), (1,3), (3,1), (3,3).