Relative Thom spectra via operadic Kan extensions

Abstract

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