Basic cohomology group decomposition of K-contact 5-manifolds

TL;DR

This paper investigates the decomposition of basic cohomology groups in 5-dimensional K-contact manifolds, establishing properties like pureness and fullness, and draws analogies with complex structures on almost complex manifolds.

Contribution

It provides a novel decomposition of basic cohomology in K-contact 5-manifolds and explores its properties, extending ideas from almost complex geometry.

Findings

Pureness and fullness of $\

Decomposition of basic degree 2 cohomology groups.

Analogy with $C^{ abla}$-pureness and fullness in almost complex manifolds.

Abstract

In this paper, we consider decompositions of basic degree 2 cohomology for a compact K-contact 5-manifold $(M,\xi,\eta,\Phi,g)$, and conclude the pureness and fullness of $\Phi$-invariant and $\Phi$-anti-invariant cohomology groups. Moreover, we discuss the decomposition of the complexified basic degree 2 cohomology group. This is an analogue problem when Draghici, Li and Zhang \cite{DLZ1} considered the $C^{\infty}$ pureness and fullness of $J$-invariant and $J$-anti-invariant subgroups of the degree 2 real cohomology group $H^2(M,\mathbb{R})$ of any compact almost complex manifold $(M, J)$.