# Enchiridion: Vol. 3 (2017): Jon Beardsley

Author
Jon Beardsley (University of Washington)

Source Paper
Relative Thom spectra via operadic Kan extensions
(arxiv:1601.04123)

Source Paper Abstract
We show that a large number of Thom spectra, i.e. colimits of morphisms $BG \rightarrow BGL_1(\mathbb{S})$, can be obtained as iterated Thom spectra, i.e. colimits of morphisms $BG \rightarrow BGL_1(Mf)$ 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.