Cohomology of measurable laminations

TL;DR

This paper introduces a novel cohomology theory for MT-spaces, including measurable foliations, extending classical algebraic topology concepts and proposing a new singular L2-cohomology applicable to these spaces.

Contribution

It develops a new cohomology framework for MT-spaces, generalizing existing theories and incorporating L2-cohomology, with applications to foliation theory.

Findings

Defines a new cohomology for MT-spaces.

Extends measurable simplicial cohomology to singular settings.

Introduces a singular L2-cohomology for MT-spaces.

Abstract

A new notion of cohomology is introduced for MT-spaces, which are measurable and topological spaces whose measurable structure may not agree with the Borel $\sigma$-algebra of their topology. The main examples of MTspaces are measurable foliations. This is a singular version of the measurable simplicial cohomology defined by Heitsch and Lazarov for foliations and extended by Bermudez for MT-spaces. Basic topics of algebraic topology are adapted, and applications to the theory of foliations are given. Moreover we introduce a new notion of singular L2-cohomology for MT-spaces.