Foundations of geometric cohomology: from co-orientations to product structures

TL;DR

This paper introduces a geometric cohomology model for smooth manifolds using co-oriented maps, emphasizing the pull-back product to realize the cup product, and unifies homology and cohomology geometrically.

Contribution

It develops a new geometric cochain complex model based on co-oriented maps and explores the partial product structure that induces the cup product in cohomology.

Findings

Constructed a geometric cochain complex model for cohomology.

Analyzed the pull-back product of smooth maps and its relation to the cup product.

Unified homology and cohomology theories geometrically.

Abstract

This manuscript develops a geometric approach to ordinary cohomology of smooth manifolds, constructing a cochain complex model based on co-oriented smooth maps from manifolds with corners. Special attention is given to the pull-back product of such smooth maps, which provides our geometric cochains with a partially defined product structure inducing the cup product in cohomology. A parallel treatment of homology is also given allowing for a geometric unification of the contravariant and covariant theories.