Geometric homology revisited

TL;DR

This paper revisits geometric homology, proposing a natural variant that replaces vector bundle modification with the Gysin map, and proves their equivalence, enhancing the geometric understanding of homology theories.

Contribution

It introduces a more natural geometric construction of homology using the Gysin map, simplifying and clarifying the relationship with cohomology theories.

Findings

Proves the equivalence of the new and classical geometric homology constructions.

Provides a natural variant of the geometric homology construction.

Enhances the geometric interpretation of homology theories.

Abstract

Given a cohomology theory, there is a well-known abstract way to define the dual homology theory using the theory of spectra. In [4] the author provides a more geometric construction of the homology theory, using a generalization of the bordism groups. Such a generalization involves in its definition the vector bundle modification, which is a particular case of the Gysin map. In this paper we provide a more natural variant of that construction, which replaces the vector bundle modification with the Gysin map itself, which is the natural push-forward in cohomology. We prove that the two constructions are equivalent.