Relative Thom Spectra Via Operadic Kan Extensions

Abstract

We show that a large number of Thom spectra, i.e. colimits of morphisms $BG\to BG{L}_1(\mathbb{S})$, can be obtained as iterated Thom spectra, i.e. colimits of morphisms $BG\to BG{L}_1(Mf)$ for some Thom spectrum $Mf$. This leads to a number of new relative Thom isomorphisms, e.g. $MU[6,\infty ){\wedge }_{MString}MU[6,\infty )\simeq MU[6,\infty )\wedge \mathbb{S}[B^{3}Spin]$. As an example of interest to chromatic homotopy theorists, we also show that Ravenel's $X(n)$ filtration of $MU$ is a tower of intermediate Thom spectra determined by a natural filtration of $BU$ by sub-bialagebras.

Repaired an error in the proof of the main theorem. This necessitated changes in Proposition 8 and Lemma 6

References in corpus (1)

• An $\infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology

Cited by in corpus (8)

• The Enriched Grothendieck Construction
• Bordism for the 2-group symmetries of the heterotic and CHL strings
• The factorization theory of Thom spectra and twisted non-abelian Poincaré duality
• A Theorem on Multiplicative Cell Attachments with an Application to Ravenel's X(n) Spectra
• The Smith Fiber Sequence and Invertible Field Theories
• Higher chromatic Thom spectra via unstable homotopy theory
• Characters and transfer maps via categorified traces
• Toward a Galois theory of the integers over the sphere spectrum