Download this article
 Download this article For screen
For printing
Recent Issues

Volume 24, 1 issue

Volume 23, 9 issues

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Author Index
To Appear
 
Other MSP Journals
Higher chromatic Thom spectra via unstable homotopy theory

Sanath K Devalapurkar

Algebraic & Geometric Topology 24 (2024) 49–108
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(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.

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

Open Access made possible by participating institutions via Subscribe to Open.