Chern numbers of almost complex manifolds (Q2750875)

It is proved that any system of numbers, that can be realized as a system of Chern numbers of an almost complex manifold of dimension \(2n\), \(n\geq 2\), can also be realized as the same system of Chern numbers of a connected almost complex manifold. This result answers an old question posed by \textit{F. Hirzebruch} [Proc. Int. Congr. Math. 1958, 119-136 (1960; Zbl 0132.19704)]. For the proof the author establishes the following lemma. Let \(M_1,\dots, M_k\) denote connected almost complex manifolds of dimension \(2n\), \(n\geq 2\). Then the connected sum \(W= M_1\sharp\cdots\sharp M_k\sharp(k- 1) S^2\times S^{2n-2}\) admits an almost complex structure, such that for any partition \(I\) of \(n\) all Chern numbers \(c_I\) satisfy the equality \(c_I(W)= c_I(M_1)+\cdots+ c_I(M_k)= c_I(M_1\sqcup\cdots\sqcup M_k)\), where \(\sqcup\) denotes disjoint union. A result of \textit{P. J. Kahn} [Ill. J. Math. 13, 336-357 (1969; Zbl 0172.25201)] is used to show the existence of the almost complex structure on \(W\).