Relative Thom Spectra Via Operadic Kan Extensions

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}$[B^3Spin]$. 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.