- Thursday Seminar Spring 2021.
Preprints and publications
Redshift and multiplication for truncated Brown-Peterson spectra
with Dylan Wilson (2020).
We equip BP<n> with an E3-BP-algebra structure, for each prime p and height n. The algebraic K-theory of this E3-ring is of chromatic height exactly n+1. Specifically, it is an fp-spectrum of fp-type n+1, which can be viewed as a higher height version of the Lichtenbaum-Quillen conjecture.
Inertia groups in the metastable range
with Robert Burklund and Andrew Senger (2020).
We prove that the inertia groups of all sufficiently-connected, high-dimensional (2n)-manifolds are trivial.
Specifically, for m≫0 and k>5/12, suppose M is a (km)-connected, smooth, closed, oriented m-manifold and Σ is an exotic m-sphere.
We prove that, if M#Σ is diffeomorphic to M, then Σ bounds a parallelizable manifold.
Our proof is an application of higher algebra in Pstragowski's category of synthetic spectra, and builds on previous work of the authors.
Galois reconstruction of Artin-Tate R-motivic spectra
(Still rough in some places) with Robert Burklund and Andrew Senger (2020).
We explain how to reconstruct the category of Artin-Tate R-motivic spectra as a deformation of the purely topological C2-equivariant stable category. The special fiber of this deformation is algebraic, and equivalent to an appropriate category of C2-equivariant sheaves on the moduli stack of formal groups. As such, our results directly generalize the cofiber of τ philosophy that has revolutionized classical stable homotopy theory.
A key observation is that the Artin-Tate subcategory of R-motivic spectra is easier to understand than the previously studied cellular subcategory. In particular, the Artin-Tate category contains a variant of the τ map, which is a feature conspicuously absent from the cellular category.
Odd primary analogs of Real orientations
with Andrew Senger and Dylan Wilson (2020).
We define, in Cp-equivariant homotopy theory for p>2, a notion of µp-orientation analogous to a C2-equivariant Real orientation. The definition hinges on a Cp-space CPµp, which we prove to be homologically even in a sense generalizing recent C2-equivariant work on conjugation spaces.
We prove that the height p-1 Morava E-theory is µp-oriented and that tmf(2) is µ3-oriented. We explain how a single equivariant map S2ρ → Σ∞CPµp completely generates the homotopy of Ep-1 and tmf(2), expressing a height-shifting phenomenon pervasive in equivariant chromatic homotopy theory.
Real topological Hochschild homology and the Segal conjecture
with Dylan Wilson (2019).
We give a new proof, independent of Lin's theorem, of the Segal conjecture for the cyclic group of order two. The key input is a calculation, as a Hopf algebroid, of the Real topological Hochschild homology of F2. This determines the E2-page of the descent spectral sequence for the map NF2 → F2, where NF2 is the C2-equivariant Hill-Hopkins-Ravenel norm of F2. The E2-page represents a new upper bound on the RO(C2)-graded homotopy of NF2, from which the Segal conjecture is an immediate corollary.
On the boundaries of highly connected, almost closed manifolds
with Robert Burklund and Andrew Senger (2019).
Building on work of Stolz, we prove for integers 0 ≤ d ≤ 3 and k > 232 that the boundaries of (k-1)-connected, almost closed (2k+d)-manifolds also bound parallelizable manifolds. Away from finitely many dimensions, this settles longstanding questions of C.T.C. Wall, determines all Stein fillable homotopy spheres, and proves a conjecture of Galatius and Randal-Williams. Implications are drawn for both the classification of highly connected manifolds and, via work of Krannich and Kreck, the calculation of their mapping class groups.
Our technique is to recast the Galatius and Randal-Williams conjecture in terms of the vanishing of a certain Toda bracket, and then to analyze this Toda bracket by bounding its HFp-Adams filtrations for all primes p. We additionally prove new vanishing lines in the HFp-Adams spectral sequences of spheres and Moore spectra, which are likely to be of independent interest. Several of these vanishing lines rely on an Appendix by Robert Burklund, which answers a question of Mathew about vanishing curves in BP<n>-based Adams spectral sequences.
Wilson Spaces, Snaith Constructions, and Elliptic Orientations
with Hood Chatham and Allen Yuan (2019).
We construct a canonical family of even periodic E∞-ring spectra, with exactly one member of the family for every prime p and chromatic height n. At height 1 our construction is due to Snaith, who built complex K-theory from CP∞. At height 2 we replace CP∞ with a p-local retract of BU<6>, producing a new theory that orients elliptic, but not generic, height 2 Morava E-theories.
In general our construction exhibits a kind of redshift, whereby BP<n-1> is used to produce a height n theory. A familiar sequence of Bocksteins, studied by Tamanoi, Ravenel, Wilson, and Yagita, relates the K(n)-localization of our height n ring to work of Peterson and Westerland building (En)hSG± from K(Z,n+1).
The Lubin-Tate Theory of Configuration Spaces: I
with Lukas Brantner and Ben Knudsen (2019).
We construct a spectral sequence converging to the Morava E-theory of unordered configuration spaces and identify its E2-page as the homology of a Chevalley-Eilenberg-like complex for Hecke Lie algebras. Based on this, we compute the E-theory of the weight p summands of iterated loop spaces of spheres (parametrising the weight p operations on En-algebras), as well as the E-theory of the configuration spaces of p points on a punctured surface. We read off the corresponding Morava K-theory groups, which appear in a conjecture by Ravenel. Finally, we compute the Fp-homology of the space of unordered configurations of p particles on a punctured surface.
Nilpotence in normed MGL-modules
with Tom Bachmann (2019).
We establish a motivic version of the May Nilpotence Conjecture: if E is a normed motivic spectrum that satisfies E^HZ=0, then also E^MGL=0. In words, motivic homology detects vanishing of normed modules over the algebraic cobordism spectrum.
Exotic Multiplications on Periodic Complex Bordism
with Allen Yuan (2019).
Victor Snaith gave a construction of periodic complex bordism by
inverting the Bott element in the suspension spectrum of BU. This
presents an E∞ structure on periodic complex bordism by different means than the usual Thom spectrum definition of the E∞-ring MUP. Here, we prove that these two E∞-rings are in fact different, though the underlying E2-rings are equivalent.
Nonetheless, we prove that both rings E∞-orient 2-completed KU and other forms of K-theory. Accepted by Journal of Topology.
Eilenberg-MacLane spectra as equivariant Thom spectra
with Dylan Wilson (2018).
We prove that the G-equivariant mod p Eilenberg-MacLane spectrum arises
as an equivariant Thom spectrum for any finite, p-power cyclic group G,
generalizing a result of Behrens and the second author in the case of
the group C2. We also establish a construction of HZ(p),
and prove intermediate results that may be of independent interest.
Highlights include constraints on the Hurewicz images of equivariant
spectra that admit norms, and an analysis of the extent to which the
non-equivariant HFp arises as the Thom spectrum of a more than double loop map. Accepted by Geometry and Topology.
Multiplicative Structure in the Stable Splitting of ΩSLn(C) with Allen Yuan (2017).
The space of based loops in SLn(C), also known as the affine Grassmannian of SLn(C), admits an E2
or fusion product. Work of Mitchell and Richter proves that this based
loop space stably splits as an infinite wedge sum. We prove that the
Mitchell--Richter splitting is coherently multiplicative, but not E2. Nonetheless, we show that the splitting becomes E2 after base-change to complex cobordism. Our proof of the A∞
splitting involves on the one hand an analysis of the multiplicative
properties of Weiss calculus, and on the other a use of
Beilinson--Drinfeld Grassmannians to verify a conjecture of Mahowald and
Richter. Other results are obtained by explicit, obstruction-theoretic
computations. Accepted by Advances in Mathematics.
Real Orientations of Lubin-Tate Theories
with Xiaolin Danny Shi (2017).
We show that Morava E-theories at the
prime 2 are Real oriented and Real Landweber exact. The proof is an
application of the Goerss-Hopkins-Miller theorem to algebras with
involution. For each height n, we compute the entire homotopy fixed
point spectral sequence for En with its C2-action
by the formal inverse. We study, as the height varies, the Hurewicz
images of the stable homotopy groups of spheres in the homotopy of these
C2-fixed points. Accepted by Inventiones Mathematicae.
Nilpotence in En algebras (2017).
Nilpotence in the homotopy of E∞-ring spectra is detected by
the classical HZ-Hurewicz homomorphism. Inspired by questions of Mathew,
Noel, and Naumann, we investigate the extent to which this criterion
holds in the homotopy of En-ring spectra. For all odd primes p
and all chromatic heights h, we use the Cohen-Moore-Neisendorfer
theorem to construct examples of K(h)-local, E2n-1-algebras with non-nilpotent pn-torsion.
We exploit the interaction of the Bousfield-Kuhn functor on odd spheres
and Rezk's logarithm to show that our bound is sharp at height 1, and
remark on the situation at height 2.
On the Bousfield classes of H∞-ring spectra (2016).
We prove that any K(n)-acyclic, H∞-ring spectrum is K(n+1)-acyclic, affirming an old conjecture of Mark Hovey.
Appendix to: Equivariant nonabelian Poincaré duality and equivariant factorization homology of Thom spectra by Asaf Horev, Inbar Klang, and Foling Zou (2020).
In an appendix joint with Dylan Wilson, we explain how to use the main theorem of the paper to assemble an integral calculation of the Real THH of Z.
Appendix to: Brown Peterson cohomology from Morava E-theory
by Tobias Barthel and Nathaniel Stapleton (2015).
We provide the proof of a technical lemma about torsion in the Morava E
theory of finite abelian groups modulo transfers. The theorems of
Barthel and Stapleton then allow conclusions about BP cohomology modulo
transfers. Accepted by Compositio Mathematica.
Quotients of even rings with Dylan Wilson (2018).
A short note about quotients of even rings remaining highly structured. Applications to Brown-Peterson spectra are forthcoming.
Work from 2013-2018 was supported by an NSF GRFP under Grant
DGE-1144152. Work starting in Fall 2018 is supported by
the NSF under Grant DMS-1803273.
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Collaborators: Tom Bachmann, Tobias Barthel, Lukas Brantner, Robert Burklund, Hood Chatham, Gijs Heuts,
Asaf Horev, Inbar Klang, Ben Knudsen, Piotr Pstragowski,
Andrew Senger, Xiaolin (Danny) Shi, Nathaniel Stapleton, Dylan Wilson, Allen Yuan, Foling Zou.