Volume 17, issue 2 (2017)

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Relative Thom spectra via operadic Kan extensions

Jonathan Beardsley

Algebraic & Geometric Topology 17 (2017) 1151–1162
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.

An errata was posted on 26 May 2017 in an online supplement.
Keywords
Thom spectra, infinity category, cobordism, cobordism spectra
Mathematical Subject Classification 2010
Primary: 55N22, 55P42
Supplementary material

Errata posted on 26 May 2017