Umkehr Maps

TL;DR

This paper provides an axiomatic framework for umkehr maps in generalized (co)homology theories, characterizing their properties and extending the theory to fiberwise contexts, with applications in string topology.

Contribution

It develops axioms that uniquely characterize umkehr homomorphisms and extends the theory to fiberwise settings inspired by string topology.

Findings

Axioms fully characterize umkehr homomorphisms

Uniqueness of umkehr homomorphisms established

Extension to fiberwise setting motivated by string topology

Abstract

In this note we study umkehr maps in generalized (co)homology theories arising from the Pontrjagin-Thom construction, from integrating along fibers, pushforward homomorphisms, and other similar constructions. We consider the basic properties of these constructions and develop axioms which any umkehr homomorphism must satisfy. We use a version of Brown representability to show that these axioms completely characterize these homomorphisms, and a resulting uniqueness theorem follows. Finally, motivated by constructions in string topology, we extend this axiomatic treatment of umkehr homomorphisms to a fiberwise setting.