Uniqueness of smooth extensions of generalized cohomology theories

TL;DR

This paper establishes an axiomatic framework demonstrating the uniqueness of smooth extensions for generalized cohomology theories, including K-theory and MU-cobordism, and relates flat theories to R/Z-cohomology.

Contribution

It introduces a set of natural axioms ensuring the uniqueness of smooth extensions and identifies flat theories with R/Z-cohomology, providing clarity on their structure.

Findings

Unique smooth extension of K-theory and MU-cobordism with multiplication

Flat theories are isomorphic to R/Z-cohomology theories

Axiomatic framework guarantees uniqueness under natural conditions

Abstract

We provide an axiomatic framework for the study of smooth extensions of generalized cohomology theories. Our main results are about the uniqeness of smooth extensions, and the identification of the flat theory with the R/Z-theory. In particular, we show that there is a unique smooth extension of K-theory and of MU-cobordism with a unique multiplication, and that the flat theory in these cases is naturally isomorphic to the homotopy theorist's version of the cohomology theory with R/Z-coefficients. For this we only require a small set of natural compatibility conditions.