Preprints
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A note on the Segal conjecture for large objects
with Vignesh Subramanian
[arxiv]
The Segal conjecture for C_p (as proved by Lin and Gunawardena) asserts that the canonical map from the p-complete sphere spectrum to the Tate construction for the trivial action of C_p on the p-complete sphere spectrum is an isomorphism. In this article we extend the collection of spectra for which the canonical map from X to X^tCp is known to be an isomorphism to include any p-complete, bounded below spectrum whose mod p homology, viewed a module over the Steenrod algebra, is complete with respect to the maximal ideal I in A.
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E_infty-coalgebras and p-adic homotopy theory
with Tom Bachmann
[arxiv]
We show that for any separably closed field k of characteristic p>0, the canonical functor from nilpotent p-adic spaces to E_infty-coalgebras over k (given by singular chains with coefficients in k) is fully faithful. We also identify the essential image of simply connected spaces inside coalgebras. This dualizes and removes finiteness assumptions from a theorem of Mandell.
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K-theoretic counterexamples to Ravenel's telescope conjecture
with Ishan Levy, Jeremy Hahn and Tomer Schlank
[arxiv]
At each prime p and height n+1≥2, we prove that the telescopic and chromatic localizations of spectra differ. Specifically, for Z acting by Adams operations on BP<n>, we prove that the T(n+1)-localized algebraic K-theory of BP<n>^hZ is not K(n+1)-local. We also show that Galois hyperdescent, A1-invariance, and nil-invariance fail for the K(n+1)-localized algebraic K-theory of K(n)-local E_infy-rings. In the case n=1 and p≥7 we make complete computations of T(2)_*K(R), for R certain finite Galois extensions of the K(1)-local sphere. We show for p≥5 that the algebraic K-theory of the K(1)-local sphere is asymptotically L2^f-local.
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Quivers and the Adams spectral sequence
with Piotr Pstragowski
[arxiv]
In this paper, we describe a novel way of identifying Adams spectral sequence E_2-terms in terms of homological algebra of quiver representations. Our method applies much more broadly than the standard techniques based on descent-flatness, bearing on a varied array of ring spectra. In the particular case of p-local integral homology, we are able to give a decomposition of the E_2-term, describing it completely in terms of the classical Adams spectral sequence. In the appendix, which can be read independently from the main body of the text, we develop functoriality of deformations of infinity-categories of the second author and Patchkoria.
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The Adams differentials on the classes h_j^3
with Zhouli Xu
[arxiv]
In filtration 1 of the Adams spectral sequence, using secondary cohomology operations, Adams computed the differentials on the classes h_j, resolving the Hopf invariant one problem. In Adams filtration 2, using equivariant and chromatic homotopy theory, Hill--Hopkins--Ravenel proved that the classes h_j^2 support non-trivial differentials for j ≥ 7, resolving the celebrated Kervaire invariant one problem. The precise differentials on the classes h_j^2 for j ≥ 7 and the fate of h_6^2 remains unknown. In this paper, in Adams filtration 3, we prove an infinite family of non-trivial d_4-differentials on the classes h_j^3 for j ≥ 6, confirming a conjecture of Mahowald. Our proof uses two different deformations of stable homotopy theory---C-motivic stable homotopy theory and F_2-synthetic homotopy theory---both in an essential way. Along the way, we also show that h_j^2 survives to the Adams E_5-page and that h_6^2 survives to the Adams E_9-page.
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The chromatic nullstellensatz
with Tomer Schlank and Allen Yuan
[arxiv]
We show that Lubin--Tate theories attached to algebraically closed fields are characterized among T(n)-local E_infty-rings as those that satisfy an analogue of Hilbert's Nullstellensatz. Furthermore, we show that for every T(n)-local E_infty-ring R, the collection of E_infty-ring maps from R to such Lubin--Tate theories jointly detect nilpotence. In particular, we deduce that every non-zero T(n)-local E_infty-ring R admits an E_infty-ring map to such a Lubin--Tate theory. As consequences, we construct E_infty complex orientations of algebraically closed Lubin--Tate theories, compute the strict Picard spectra of such Lubin--Tate theories, and prove redshift for the algebraic K-theory of arbitrary E_infty-rings.
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Multiplicative structures on Moore spectra
[arxiv]
In this article we show that S/8 is an E_1-algebra, S/32 is an E_2-algebra, S/p^(n+1) is an E_n-algebra at odd primes and, more generally, for every h and n there exist generalized Moore spectra of type h which admit an E_n-algebra structure.
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How big are the stable homotopy groups of spheres?
including an appendix joint with Andrew Senger
[arxiv]
In this article we show that the p-torsion exponent of the stable stems grows sublinearly in n and the p-rank of the E2-page of the Adams spectral sequence grows as exp(theta(log(n)^3)). Together these bounds provide the first subexponential bound on the size of the stable stems. Conversely, we prove that a certain, precise, version of the failure of the telescope conjecture would imply that the upper bound provided by the Adams E2-page is essentially sharp -- answering the titular question: As big as the fate of the telescope conjecture demands.
In an appendix joint with Andrew Senger we consider the unstable analog of this question. Bootstrapping from the stable bounds we prove that the size of the p-local homotopy groups of spheres grows like exp(O(log(n)^3)), providing the first subexponential bound on the size of the unstable stems.
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Galois reconstruction of Artin-Tate R-motivic spectra
with Jeremy Hahn and Andrew Senger
[arxiv]
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 tau 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 tau map, which is a feature conspicuously absent from the cellular category.
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Inertia groups in the metastable range
with Jeremy Hahn and Andrew Senger
[arxiv]
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.
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