Higher chromatic Thom spectra via unstable homotopy theory

Devalapurkar, Sanath K¹

¹ Department of Mathematics, Harvard University, Cambridge, MA, United States

Algebraic and Geometric Topology, Tome 24 (2024) no. 1, pp. 49-108

Résumé

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.

Détail

DOI : 10.2140/agt.2024.24.49

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