Relative Thom Spectra Via Operadic Kan Extensions

TL;DR

This paper demonstrates that many Thom spectra can be constructed as iterated Thom spectra, leading to new relative Thom isomorphisms and insights into the structure of spectra relevant to chromatic homotopy theory.

Contribution

It introduces a method to obtain Thom spectra as colimits of morphisms into Thom spectra, revealing new isomorphisms and a natural filtration of Ravenel's $X(n)$ spectra.

Findings

New relative Thom isomorphisms for various spectra

Representation of Ravenel's $X(n)$ as a tower of Thom spectra

Enhanced understanding of Thom spectrum constructions

Abstract

We show that a large number of Thom spectra, i.e. colimits of morphisms $BG\to BGL_1(\mathbb{S})$, can be obtained as iterated Thom spectra, i.e. colimits of morphisms $BG\to BGL_1(Mf)$ for some Thom spectrum $Mf$. This leads to a number of new relative Thom isomorphisms, e.g. $MU[6,\infty)\wedge_{MString} MU[6,\infty)\simeq MU[6,\infty)\wedge\mathbb{S}[B^3Spin]$. 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.