English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

This paper studies existence of $n=4k (k>1)$ dimensional simply-connectedclosed 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 showthey must have even Euler characteristic and even signature, which is to saythe middle Betti number $b_{n/2}$ must be even. Parallel to the author'searlier work on realizing rational cohomology ring by smooth closed manifolds,we state and prove Sullivan's rational surgery realization theorem for almostcomplex manifold and demonstrate its application in our context. A prescribedrational cohomology ring can be realized by a simply connected almost complexmanifold if and only if the ring structure supports the intersection form of aclosed manifold, and it holds Chern numbers that satisfy the signature equationand the Riemann-Roch integrality relations, and the top Chern number equals theEuler characteristic. According to Stong's characterization of $U$ and $SU$cobordism, we explicitly compute the Riemann-Roch integrality relations amongChern numbers in the case when only the middle and top Chern classes can benonzero. The necessary and sufficient conditions for realization are expressedas 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 theEuler characteristic increase with respect to the dimension of the realizingmanifold.

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