Multiplicative Parametrized Homotopy Theory via Symmetric Spectra in Retractive Spaces

Abstract

In order to treat multiplicative phenomena in twisted (co)homology, we introduce a new point-set level framework for parametrized homotopy theory. We provide a convolution smash product that descends to the corresponding ∞-categorical product and allows for convenient constructions of commutative parametrized ring spectra. As an immediate application, we compare various models for generalized Thom spectra. In a companion paper, this approach is used to compare homotopical and operator algebraic models for twisted K-theory.

Twisted spin cobordism and positive scalar curvature

“…Remark. As mentioned in Remark 3.1.4, based on the observations of this Appendix a more refined category of parametrised spectra is constructed in joint work of the first author with Sagave and Schlichtkrull [17]. In this set-up the construction of M Spin K and KO K can be carried out without first passing to the primed free rank 1 modules and in a second paper [16] the results of this Appendix are then strengthened to hold at the level of twisted cohomology, rather than just at the level of twists.…”

Homotopical and operator algebraic twisted K-theory

“…We now outline the definition of the parametrized spectra that we will be using throughout the paper. These are the symmetric spectra in retractive spaces developed in a companion paper joint with Schlichtkrull [15].…”

“…A central feature of I-spaces is that they allow to model E ∞ spaces by commutative I-space monoids, i.e., by strictly commutative monoids with respect to [22,Theorem 1.2]. If M is a commutative I-space monoid, we show that the category Sp Σ M inherits a well-behaved symmetric monoidal structure, the convolution smash product [15,Section 4]. To our knowledge, this product is not present in other point-set approaches to parametrized homotopy theory.…”

“…The extensive comparison results of [15,Section 10] allow us to obtain the statement of the theorem also in the ∞-categorical models for parametrized spectra considered in [2,3]. To an E ∞ -space the authors of these papers associate a symmetric monoidal ∞-category of parametrized spectra, which is equivalent to the underlying ∞-category of the category of parametrized spectra with the convolution smash product employed in the present paper.…”

“…In the present paper, we use foundational joint work with Schlichtkrull [15] to provide such a comparison. The main feature of the framework established in [15] is a point set level implementation of the convolution smash product of parametrized spectra that was previously only available in ∞-categorical contexts. By considering natural refinements of taut * KO (γ KO ), κ * (γ KO ) and KO PO to commutative parametrized ring spectra in this setting, we show:…”