Reduced $L_{q,p}$-Cohomology of Some Twisted Products

TL;DR

This paper proves vanishing results for reduced $L_{p,q}$-cohomology in twisted products, extending previous work on warped cylinders and showing that certain middle-dimensional cohomology groups are zero.

Contribution

It extends vanishing results of $L_{p,q}$-cohomology to twisted products, generalizing earlier results on warped cylinders and analyzing middle-dimensional cohomology.

Findings

Vanishing of reduced $L_{p,q}$-cohomology in twisted products.

Zero $L_2$-Betti numbers in middle dimension.

Extension of previous $L_{p}$-cohomology results.

Abstract

Vanishing results for reduced $L_{p,q}$-cohomology are established in the case of twisted products, which are a~generalization of warped products. Only the case $q \leq p$ is considered. This is an extension of some results by Gol'dshtein, Kuz'minov and Shvedov about the $L_{p}$-cohomology of warped cylinders. One of the main observations is the vanishing of the "middle-dimensional" cohomology for a large class of manifolds. This means that the $L_2$-Betty numbers are zero in the "middle dimension".