# Homotopy invariance of cohomology and signature of a riemannian foliation

## TL;DR

This paper establishes that the basic signature of a Riemannian foliation is a homotopy invariant and demonstrates invariance properties of related cohomological and geometric structures under foliated homotopy.

## Contribution

It proves the basic signature is a well-defined, homotopy-invariant characteristic of Riemannian foliations, extending the understanding of foliation invariants.

## Key findings

- Basic signature is a foliated homotopy invariant.
- Foliated homotopic maps induce isomorphic basic Lichnerowicz cohomology.
- Alvarez class remains invariant under foliated homotopy.

## Abstract

We prove that any smooth foliation that admits a Riemannian foliation structure has a well-defined basic signature, and this geometrically defined invariant is actually a foliated homotopy invariant. We also show that foliated homotopic maps between Riemannian foliations induce isomorphic maps on basic Lichnerowicz cohomology, and that the Alvarez class of a Riemannian foliation is invariant under foliated homotopy equivalence.