The cotangent complex and Thom spectra

The cotangent complex of a map of commutative rings is a central object in deformation theory. Since the 1990s, it has been generalized to the homotopical setting of $$E_\infty $$ E ∞ -ring spectra in various ways. In this work we first establish, in the context of $$\infty $$ ∞ -categories and using Goodwillie’s calculus of functors, that various definitions of the cotangent complex of a map of $$E_\infty $$ E ∞ -ring spectra that exist in the literature are equivalent. We then turn our attention to a specific example. Let R be an $$E_\infty $$ E ∞ -ring spectrum and $$\mathrm {Pic}(R)$$ Pic ( R ) denote its Picard $$E_\infty $$ E ∞ -group. Let Mf denote the Thom $$E_\infty $$ E ∞ -R-algebra of a map of $$E_\infty $$ E ∞ -groups $$f:G\rightarrow \mathrm {Pic}(R)$$ f : G → Pic ( R ) ; examples of Mf are given by various flavors of cobordism spectra. We prove that the cotangent complex of $$R\rightarrow Mf$$ R → M f is equivalent to the smash product of Mf and the connective spectrum associated to G.

MULTIPLICATIVE PARAMETRIZED HOMOTOPY THEORY VIA SYMMETRIC SPECTRA IN RETRACTIVE SPACES

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 $\infty$ -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.

THE UNIT MAP OF THE ALGEBRAIC SPECIAL LINEAR COBORDISM SPECTRUM

In the joint work with Elmanto, Hoyois, Khan and Sosnilo, we computed infinite $\mathbb{P}^{1}$ -loop spaces of motivic Thom spectra using the technique of framed correspondences. This result allows us to express non-negative $\mathbb{G}_{m}$ -homotopy groups of motivic Thom spectra in terms of geometric generators and relations. Using this explicit description, we show that the unit map of the algebraic special linear cobordism spectrum induces an isomorphism on $\mathbb{G}_{m}$ -homotopy sheaves.

GENERALIZED THOM SPECTRA AND THEIR TOPOLOGICAL HOCHSCHILD HOMOLOGY

We develop a theory of $R$-module Thom spectra for a commutative symmetric ring spectrum $R$ and we analyze their multiplicative properties. As an interesting source of examples, we show that $R$-algebra Thom spectra associated to the special unitary groups can be described in terms of quotient constructions on $R$. We apply the general theory to obtain a description of the $R$-based topological Hochschild homology associated to an $R$-algebra Thom spectrum.