Cohomology of manifolds with structure group $U(n)\times O(s)$

TL;DR

This paper develops a spectral sequence approach to study the cohomology of manifolds with structure group $U(n) imes O(s)$, generalizing results from $K$-contact geometry to $ ext{S}$- and $ ext{C}$-manifolds, and establishing invariance properties of their cohomology.

Contribution

It introduces a new spectral sequence for $ ext{K}$-manifolds, computes cohomology rings and harmonic forms for $ ext{S}$-manifolds, and proves invariance of basic cohomology and Hodge numbers under deformations.

Findings

Cohomology ring of $ ext{S}$-manifolds expressed via primitive basic cohomology.

Basic cohomology of $ ext{S}$-manifolds is a topological invariant.

Basic Hodge numbers are invariant under deformations.

Abstract

We introduce a new spectral sequence for the study of $\mathcal{K}$-manifolds which arises by restricting the spectral sequence of a Riemannian foliation to forms invariant under the flows of $\{\xi_1,...,\xi_s\}$. We use this sequence to generalize a number of theorems from $K$-contact geometry to $\mathcal{K}$-manifolds. Most importantly we compute the cohomology ring and harmonic forms of $\mathcal{S}$-manifolds in terms of primitive basic cohomology and primitive basic harmonic forms (respectively). As an immediate consequence of this we get that the basic cohomology of $\mathcal{S}$-manifolds are a topological invariant. We also show that the basic Hodge numbers of $\mathcal{S}$-manifolds are invariant under deformations. Finally, we provide similar results for $\mathcal{C}$-manifolds.