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Higher chromatic Thom spectra via unstable homotopy theory

Sanath K Devalapurkar

Abstract

We investigate implications of an old conjecture in unstable homotopy theory related to the Cohen-Moore-Neisendorfer theorem and a conjecture about the $\mathbf{E}_{2}$-topological Hochschild cohomology of certain Thom spectra (denoted $A$, $B$, and $T(n)$) related to Ravenel's $X(p^n)$. We show that these conjectures imply that the orientations $\mathrm{MSpin}\to \mathrm{ko}$ and $\mathrm{MString}\to \mathrm{tmf}$ admit spectrum-level splittings. This is shown by generalizing a theorem of Hopkins and Mahowald, which constructs $\mathrm{H}\mathbf{F}_p$ as a Thom spectrum, to construct $\mathrm{BP}\langle{n-1}\rangle$, $\mathrm{ko}$, and $\mathrm{tmf}$ as Thom spectra (albeit over $T(n)$, $A$, and $B$ respectively, and not over the sphere). This interpretation of $\mathrm{BP}\langle{n-1}\rangle$, $\mathrm{ko}$, and $\mathrm{tmf}$ offers a new perspective on Wood equivalences of the form $\mathrm{bo} \wedge Cη\simeq \mathrm{bu}$: they are related to the existence of certain EHP sequences in unstable homotopy theory. This construction of $\mathrm{BP}\langle{n-1}\rangle$ also provides a different lens on the nilpotence theorem. Finally, we prove a $C_2$-equivariant analogue of our construction, describing $\underline{\mathrm{H}\mathbf{Z}}$ as a Thom spectrum.

Higher chromatic Thom spectra via unstable homotopy theory

Abstract

We investigate implications of an old conjecture in unstable homotopy theory related to the Cohen-Moore-Neisendorfer theorem and a conjecture about the -topological Hochschild cohomology of certain Thom spectra (denoted , , and ) related to Ravenel's . We show that these conjectures imply that the orientations and admit spectrum-level splittings. This is shown by generalizing a theorem of Hopkins and Mahowald, which constructs as a Thom spectrum, to construct , , and as Thom spectra (albeit over , , and respectively, and not over the sphere). This interpretation of , , and offers a new perspective on Wood equivalences of the form : they are related to the existence of certain EHP sequences in unstable homotopy theory. This construction of also provides a different lens on the nilpotence theorem. Finally, we prove a -equivariant analogue of our construction, describing as a Thom spectrum.

Paper Structure

This paper contains 27 sections, 50 theorems, 93 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. The statements of Conjectures \ref{['moore-splitting']}, \ref{['centrality-conj']}, and \ref{['tmf-conj']}
  4. Outline
  5. Background, and some classical positive and negative results
  6. Background on Thom spectra
  7. The Hopkins-Mahowald theorem
  8. No-go theorems for higher chromatic heights
  9. Some Thom spectra
  10. Ravenel's $X(n)$ spectra
  11. Related Thom spectra
  12. Centers of Thom spectra
  13. Review of some unstable homotopy theory
  14. Charming and Gray maps
  15. Fibers of charming maps
  16. ...and 12 more sections

Key Result

Theorem 1

Let $\mu:\Omega^2 S^3\to \mathrm{BO}$ denote the real vector bundle over $\Omega^2 S^3$ induced by extending the map $S^1\to \mathrm{BO}$ classifying the Möbius bundle. Then the Thom spectrum of $\mu$ is equivalent to $\mathrm{H}\mathbf{F}_2$ as an $\mathbf{E}_{{2}}$-algebra.

Figures (3)

  • Figure 1: $C\eta\wedge C\nu$ shown horizontally, with $0$-cell on the left. The element $\sigma_1$ is given by the map $\eta$ on the $4$-cell defined by a nullhomotopy of $\eta\nu = 0\in \pi_4(S^0)$, as indicated in the diagram above.
  • Figure 2: $15$-skeleton of $A$ at the prime $2$ shown horizontally, with $0$-cell on the left. The element $\sigma_1$ given by the map $\eta$ on the $4$-cell, as indicated in the diagram above.
  • Figure 3: Steenrod module structure of the $20$-skeleton of $B$; the bottom cell (in dimension $0$) is on the left; straight lines are $\mathrm{Sq}^4$, and curved lines correspond to $\mathrm{Sq}^8$ and $\mathrm{Sq}^{16}$, in order of increasing length. The bottom two attaching maps of $B$ are labeled. The map $\sigma_2$ is shown.

Theorems & Definitions (168)

  • Conjecture 1.1.1
  • Theorem : Hopkins-Mahowald; see mahowald-thom and mrs
  • Remark 1.1.2
  • Conjecture 1.1.3: Hahn, Hopkins; unpublished
  • Conjecture 1.1.4: Hahn
  • Theorem A
  • Corollary B
  • Theorem C
  • Conjecture D
  • Conjecture E
  • ...and 158 more