Commutativity properties of Quinn spectra

TL;DR

This paper establishes conditions under which Quinn spectra are equivalent to commutative symmetric ring spectra and explores the monoidal properties of the symmetric signature, advancing the understanding of algebraic topology structures.

Contribution

It provides a simple sufficient condition for Quinn's spectra to be commutative and shows the symmetric signature as a monoidal transformation, linking topological bundles to ring spectra.

Findings

Quinn spectra can be weakly equivalent to commutative symmetric ring spectra under certain conditions.

The symmetric signature acts as a monoidal transformation between symmetric monoidal functors.

A new description of symmetric L theory is introduced, potentially of independent interest.

Abstract

We give a simple sufficient condition for Quinn's "bordism-type" spectra to be weakly equivalent to commutative symmetric ring spectra. We also show that the symmetric signature is (up to weak equivalence) a monoidal transformation between symmetric monoidal functors, which implies that the Sullivan-Ranicki orientation of topological bundles is represented by a ring map between commutative symmetric ring spectra. In the course of proving these statements we give a new description of symmetric L theory which may be of independent interest.