Geometric Homology

TL;DR

This paper develops a new version of singular homology using smooth mappings of manifolds with corners, emphasizing transversality and intersection theory without relying on triangulation, to improve conceptual clarity and computational simplicity.

Contribution

It introduces a chain level homology theory based on smooth manifolds with corners, avoiding triangulation and highlighting intersection theory for better intuition.

Findings

Constructed a chain level theory based on smooth mappings.

Avoided using triangulation of manifolds.

Facilitated simpler computations through intersection theory.

Abstract

The purpose of this paper is to introduce a version of singular homology based on smooth mappings of manifolds with corners. Although variants of such a theory exists in the literature, we felt that certain points were not adequately addressed. In particular, our goal is to construct a chain level theory based on smooth mappings of manifolds with corners. In addition, we will avoid using the fact that smooth manifolds with can be triangulated. As we shall see, transversality and intersections play a major role in setting up this theory. From a pedagogical viewpoint, having intersection theory arguments available from the start facilitates simple and intuitive computations.