Transgressions and Chern characters in coarse homotopy theory

TL;DR

This paper explores the relationships between coarse homology theories, transgressions, and Chern characters, focusing on their natural transformations and the commutativity of related diagrams in coarse homotopy theory.

Contribution

It studies the interplay between analytical and topological transgressions and algebraic and homotopy Chern characters within coarse homology theories.

Findings

Established conditions for commutativity of the transgression-Chern character square.

Analyzed the factorization of functors over the Higson corona.

Clarified the natural transformations connecting coarse homology theories.

Abstract

This paper investigates a variety of coarse homology theories and natural transformations between them. We in particular study the commutativity of a square relating analytical and topological transgressions with algebraic and homotopy theoretic Chern characters. Here a transgression is a natural transformation from a coarse homology theory to a functor which factorizes over the Higson corona functor, and a Chern character is a transformation from a $K$-theory like coarse or Borel-Moore type homology theory to an ordinary version.