A theory of twisted umkehr maps

TL;DR

This paper develops a universal categorical framework for twisted umkehr maps in generalized homology theories, enabling systematic access to classical and new maps.

Contribution

It introduces a theory where twisted umkehr maps are induced by cartesian morphisms, generalizing classical constructions via universal properties.

Findings

Unified categorical framework for twisted umkehr maps

Includes generalizations of classical umkehr maps

Access through universal properties

Abstract

We develop a theory of umkehr maps for twisted generalized homology theories. In this theory, interesting umkehr maps, including generalizations of important classical ones, are induced by cartesian morphisms of a certain category opfibred over the category of spaces and continuous maps, making it possible to access them through universal properties.