Higher chromatic Thom spectra via unstable homotopy theory

Devalapurkar, Sanath K

doi:10.2140/agt.2024.24.49

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Higher chromatic Thom spectra via unstable homotopy theory

Sanath K Devalapurkar

Algebraic & Geometric Topology 24 (2024) 49–108

DOI: 10.2140/agt.2024.24.49

Abstract

We investigate implications of an old conjecture in unstable homotopy theory related to the Cohen–Moore–Neisendorfer theorem and a conjecture about the $E [2]$ –topological Hochschild cohomology of certain Thom spectra (denoted by $A$ , $B$ and $T (n)$ ) related to Ravenel’s $X (p^{n})$ . We show that these conjectures imply that the orientations $M Spin \to b o$ and $M String \to tmf$ admit spectrum-level splittings. This is shown by generalizing a theorem of Hopkins and Mahowald, which constructs $H F_{p}$ as a Thom spectrum, to construct $BP ⟨ n - 1 ⟩$ , $b o$ , and $tmf$ as Thom spectra (albeit over $T (n)$ , $A$ , and $B “$ , respectively, and not over the sphere). This interpretation of $BP ⟨ n - 1 ⟩$ , $b o$ , and $tmf$ offers a new perspective on Wood equivalences of the form $b o \land C η ≃ b u$ : they are related to the existence of certain EHP sequences in unstable homotopy theory. This construction of $BP ⟨ n - 1 ⟩$ also provides a different lens on the nilpotence theorem. Finally, we prove a $C_{2}$ –equivariant analogue of our construction, describing $\underline{H Z}$ as a Thom spectrum.

Keywords

Thom spectra, chromatic homotopy theory, Steenrod algebra, unstable homotopy theory

Mathematical Subject Classification

Primary: 55N34, 55P43, 55S12

References

Publication

Received: 14 November 2020

Revised: 22 August 2022

Accepted: 10 October 2022

Published: 18 March 2024

Authors

Sanath K Devalapurkar
Department of Mathematics Harvard University Cambridge, MA United States

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Volume 24, issue 1 (2024)