The period isomorphism in tame geometry

TL;DR

This paper introduces a novel approach to describing singular homology in tame geometry using simplices satisfying Stokes' formula, enabling a new perspective on the period pairing with de Rham cohomology in definable manifolds.

Contribution

It provides a new characterization of singular homology in tame geometry through simplices that satisfy Stokes' formula, linking it to the period pairing with de Rham cohomology.

Findings

Describes singular homology via simplices satisfying Stokes' formula.

Connects the homology description to the period pairing in tame geometry.

Applicable to definable manifolds in o-minimal structures.

Abstract

We describe singular homology of a manifold $X$ via simplices $\sigma:\Delta_d\to X$ that satisfy Stokes' formula with respect to all differential forms. The notion is geared to the case of tame geometry (definable manifolds with respect to an o-minimal structure), where it gives a description of the period pairing with de Rham cohomology via definable $\sigma$'s.