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Higher chromatic Thom spectra via unstable homotopy theory

Sanath K Devalapurkar

Algebraic & Geometric Topology 24 (2024) 49–108

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(pn). We show that these conjectures imply that the orientations MSpin bo and MString tmf admit spectrum-level splittings. This is shown by generalizing a theorem of Hopkins and Mahowald, which constructs HFp as a Thom spectrum, to construct BP n 1, bo, and tmf as Thom spectra (albeit over T(n), A, and B, respectively, and not over the sphere). This interpretation of BP n 1, bo, and tmf offers a new perspective on Wood equivalences of the form bo Cη bu: 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 C2–equivariant analogue of our construction, describing HZ ¯ as a Thom spectrum.

Thom spectra, chromatic homotopy theory, Steenrod algebra, unstable homotopy theory
Mathematical Subject Classification
Primary: 55N34, 55P43, 55S12
Received: 14 November 2020
Revised: 22 August 2022
Accepted: 10 October 2022
Published: 18 March 2024
Sanath K Devalapurkar
Department of Mathematics
Harvard University
Cambridge, MA
United States

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