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Commutativity properties of Quinn spectra

Gerd Laures, James E. McClure

TL;DR

This work establishes that Quinn spectra arising from bordism-type theories admit commutative models by constructing a commutative ad theory ${\mathbf M}^{\mathrm{comm}}$ weakly equivalent to the original ${\mathbf M}$ via an $E_{\infty}$-operad action and a bar construction. It demonstrates multiplicativity of the symmetric signature as a monoidal transformation and represents the Sullivan–Ranicki orientation as a ring map between commutative symmetric ring spectra, connecting bordism and L-theory in a strictly multiplicative framework. A key technical advance is a rectification pipeline built from a family of permutation-interpolating multiplications, organized into monads on multisemisimplicial symmetric spectra and their product variants, which yields strictly commutative models after realization. The paper also introduces relaxed symmetric Poincaré complexes to provide an alternative, yet equivalent, description of the L-spectrum, enabling clean multiplicativity arguments and a robust comparison between ad theories. Together, these results place bordism, L-theory, and related signatures in a coherent, commutative spectral framework with explicit, model-level maps and equivalences, advancing applications in surgery theory and orientation theory.

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.

Commutativity properties of Quinn spectra

TL;DR

This work establishes that Quinn spectra arising from bordism-type theories admit commutative models by constructing a commutative ad theory weakly equivalent to the original via an -operad action and a bar construction. It demonstrates multiplicativity of the symmetric signature as a monoidal transformation and represents the Sullivan–Ranicki orientation as a ring map between commutative symmetric ring spectra, connecting bordism and L-theory in a strictly multiplicative framework. A key technical advance is a rectification pipeline built from a family of permutation-interpolating multiplications, organized into monads on multisemisimplicial symmetric spectra and their product variants, which yields strictly commutative models after realization. The paper also introduces relaxed symmetric Poincaré complexes to provide an alternative, yet equivalent, description of the L-spectrum, enabling clean multiplicativity arguments and a robust comparison between ad theories. Together, these results place bordism, L-theory, and related signatures in a coherent, commutative spectral framework with explicit, model-level maps and equivalences, advancing applications in surgery theory and orientation theory.

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.

Paper Structure

This paper contains 30 sections, 65 theorems, 449 equations.

Table of Contents

  1. Introduction
  2. Some redefinitions
  3. Commutative ad theories
  4. Multisemisimplicial symmetric spectra
  5. Geometric realization
  6. A family of multiplication maps
  7. A monad in $\Sigma{{ss}}{\mathcal{S}}$
  8. A monad in ${\mathcal{S}} p_{{mss }}$
  9. Degreewise smash product and geometric realization
  10. Rectification
  11. Proof of Theorem \ref{['m96']}
  12. Relaxed symmetric Poincaré complexes
  13. Equivalence of the spectra associated to ${\mathrm{ad}}^R$ and ${\mathrm{ad}}_{\text{\rm{rel}}}^R$
  14. The symmetric signature revisited
  15. Background for the proof of Theorem \ref{['m54']}
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let ${\mathbf M}$ be the symmetric ring spectrum associated to a commutative ad theory. There is a commutative symmetric ring spectrum ${\mathbf M}^{\mathrm{comm}}$ which is weakly equivalent in the category of symmetric ring spectra to ${\mathbf M}$ and depends on it in a natural way.

Theorems & Definitions (214)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Remark 3.4
  • Definition 3.5
  • ...and 204 more