Multiplicative properties of Quinn spectra

TL;DR

This paper establishes conditions under which Quinn's bordism spectra are equivalent to associative ring spectra, and demonstrates natural equivalences among Poincare bordism, symmetric L-theory, and monoidal functors.

Contribution

It provides a simple sufficient condition for Quinn's spectra to be strictly associative and explores their relation to monoidal functors and symmetric spectra.

Findings

Quinn's bordism spectra can be weakly equivalent to strictly associative ring spectra.

Poincare bordism and symmetric L-theory are weakly equivalent to monoidal functors.

The functor from bordism theories to spectra lifts to symmetric spectra.

Abstract

We give a simple sufficient condition for Quinn's "bordism-type spectra" to be weakly equivalent to strictly associative ring spectra. We also show that Poincare bordism and symmetric L-theory are naturally weakly equivalent to monoidal functors. Part of the proof of these statements involves showing that Quinn's functor from bordism-type theories to spectra lifts to the category of symmetric spectra. We also give a new account of the foundations.