Almost complex manifold with Betti number $b_i=0$ except $i=0, n/2, n$

Abstract

This paper studies existence of $n=4k (k>1)$ dimensional simply-connected closed almost complex manifold with Betti number $ b_i=0$ except $i=0, n/2, n$. We characterize all the rational cohomology rings of such manifolds and show they must have even Euler characteristic and even signature, which is to say the middle Betti number $b_{n/2}$ must be even. Parallel to the author's earlier work on realizing rational cohomology ring by smooth closed manifolds, we state and prove Sullivan's rational surgery realization theorem for almost complex manifold and demonstrate its application in our context. A prescribed rational cohomology ring can be realized by a simply connected almost complex manifold if and only if the ring structure supports the intersection form of a closed manifold, and it holds Chern numbers that satisfy the signature equation and the Riemann-Roch integrality relations, and the top Chern number equals the Euler characteristic. According to Stong's characterization of $U$ and $SU$ cobordism, we explicitly compute the Riemann-Roch integrality relations among Chern numbers in the case when only the middle and top Chern classes can be nonzero. The necessary and sufficient conditions for realization are expressed as a set of congruence relations among the signature and Euler characteristic, we show that the lower bounds of the 2-adic order of the signature and the Euler characteristic increase with respect to the dimension of the realizing manifold.