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Jeremy Hahn

Publications and source records attributed to Jeremy Hahn.

At least 19 recordsLinked to original sources
arXiv2024

Syntomic cohomology of Morava K-theory

We compute the MU-based syntomic cohomologies, mod $(p,v_1,\cdots,v_{n+1})$, of all $\mathbb{E}_1$-MU-algebra forms of connective Morava K-theory k(n). As qualitative consequences, we deduce the Lichtenbaum--Quillen conjecture, telescope conjecture, and redshift conjecture for the algebraic K-theories of all $\mathbb{E}_{1}$-$\mathbb{S}$-algebra forms of $(2p^n-2)$-periodic Morava K-theory. Notably, the motivic spectral sequence computing $\pi_*TC(k(n))_p$ is concentrated on at most three lines, independently of $n$.

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arXiv2024

Crystallinity for syntomic cohomology, \'etale cohomology, and algebraic $K$-theory

We prove for $n\geq c-1$ that the functor taking an animated ring $R$ to its mod $(p^c,v_1^{p^n})$ syntomic cohomology factors through the functor $R \mapsto R/p^{c(n+2)}$, a phenomenon we term crystallinity for mod $(p^c,v_1^{p^n})$ syntomic cohomology. As an application, we completely and explicitly compute the mod $(p,v_1 ^{p^{n}-1})$ algebraic $K$-theory of $\mathbb Z/p^{k}$ whenever $k \geq n+2$ and $p>2$. As a second application, we deduce crystallinity for the mod $p^c$ syntomic complexes associated to smooth $p$-adic formal schemes, and in particular for the Galois equivariant mod $p^c$ \'etale cohomologies of their adic generic fibers. Finally, we strengthen known $p$-adic convergence theorems for the topological Hochschild homology of ring spectra, and as a result relate crystallinity for algebraic $K$-theory to Lichtenbaum--Quillen theorems.

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arXiv2023

$K$-theoretic counterexamples to Ravenel's telescope conjecture

At each prime $p$ and height $n+1 \ge 2$, we prove that the telescopic and chromatic localizations of spectra differ. Specifically, for $\mathbb{Z}$ acting by Adams operations on $\mathrm{BP}\langle n \rangle$, we prove that the $T(n+1)$-localized algebraic $K$-theory of $\mathrm{BP}\langle n \rangle^{h\mathbb{Z}}$ is not $K(n+1)$-local. We also show that Galois hyperdescent, $\mathbb{A}^1$-invariance, and nil-invariance fail for the $K(n+1)$-localized algebraic $K$-theory of $K(n)$-local $\mathbb{E}_{\infty}$-rings. In the case $n=1$ and $p \ge 7$ we make complete computations of $T(2)_*\mathrm{K}(R)$, for $R$ certain finite Galois extensions of the $K(1)$-local sphere. We show for $p\geq 5$ that the algebraic $K$-theory of the $K(1)$-local sphere is asymptotically $L_2^{f}$-local.

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arXiv2023

Examples of disk algebras

We produce refinements of the known multiplicative structures on the Brown--Peterson spectrum $BP$, its truncated variants $BP\langle n \rangle$, Ravenel's spectra $X(n)$, and evenly graded polynomial rings over the sphere spectrum. Consequently, topological Hochschild homology relative to these rings inherits a circle action.

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arXiv2022

Real topological Hochschild homology and the Segal conjecture

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 $\mathbb{F}_2$. This determines the $\mathrm{E}_2$-page of the descent spectral sequence for the map $\mathrm{N}\mathbb{F}_2 \to \mathbb{F}_2$, where $\mathrm{N}\mathbb{F}_2$ is the $C_2$-equivariant Hill--Hopkins--Ravenel norm of $\mathbb{F}_2$. The $\mathrm{E}_2$-page represents a new upper bound on the $RO(C_2)$-graded homotopy of $\mathrm{N}\mathbb{F}_2$, from which the Segal conjecture is an immediate corollary.

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arXiv2022

Redshift and multiplication for truncated Brown-Peterson spectra

We equip $\mathrm{BP} \langle n \rangle$ with an $\mathbb{E}_3$-$\mathrm{BP}$-algebra structure, for each prime $p$ and height $n$. The algebraic $K$-theory of this ring is of chromatic height exactly $n+1$, and the map $\mathrm{K}(\mathrm{BP}\langle n \rangle)_{(p)} \to \mathrm{L}_{n+1}^{f} \mathrm{K}(\mathrm{BP}\langle n\rangle)_{(p)}$ has bounded above fiber.

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arXiv2022

A motivic filtration on the topological cyclic homology of commutative ring spectra

For a prime number $p$ and a $p$-quasisyntomic commutative ring $R$, Bhatt--Morrow--Scholze defined motivic filtrations on the $p$-completions of $\mathrm{THH}(R), \mathrm{TC}^{-}(R), \mathrm{TP}(R),$ and $\mathrm{TC}(R)$, with the associated graded objects for $\mathrm{TP}(R)$ and $\mathrm{TC}(R)$ recovering the prismatic and syntomic cohomology of $R$, respectively. We give an alternate construction of these filtrations that applies also when $R$ is a well-behaved commutative ring spectrum; for example, we can take $R$ to be $\mathbb{S}$, $\mathrm{MU}$, $\mathrm{ku}$, $\mathrm{ko}$, or $\mathrm{tmf}$. We compute the mod $(p,v_1)$ syntomic cohomology of the Adams summand $\ell$ and observe that, when $p \ge 3$, the motivic spectral sequence for $V(1)_*\mathrm{TC}(\ell)$ collapses at the $\mathrm{E}_2$-page.

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arXiv2022

Nilpotence in normed MGL-modules

We establish a motivic version of the May Nilpotence Conjecture: if E is a normed motivic spectrum that satisfies $E \wedge HZ \simeq 0$, then also $E \wedge MGL \simeq 0$. In words, motivic homology detects vanishing of normed modules over the algebraic cobordism spectrum.

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arXiv2022

On the boundaries of highly connected, almost closed manifolds

Building on work of Stolz, we prove for integers $0 \le d \le 3$ 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 Kreck and Krannich, 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 $\mathrm{H}\mathbb{F}_p$-Adams filtrations for all primes $p$. We additionally prove new vanishing lines in the $\mathrm{H}\mathbb{F}_p$-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 $\mathrm{BP} \langle n \rangle$-based Adams spectral sequences.

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arXiv2022

Inertia groups in the metastable range

We prove that the inertia groups of all sufficiently-connected, high-dimensional $(2n)$-manifolds are trivial. This is a key step toward a general classification of manifolds in the metastable range. Specifically, for $m \gg 0$ and $k>5/12$, suppose $M$ is a $\lfloor km \rfloor$-connected, smooth, closed, oriented $m$-manifold and $Σ$ is an exotic $m$-sphere. We prove that, if $M \sharp Σ$ is diffeomorphic to $M$, then $Σ$ bounds a parallelizable manifold. Our proof is built on an understanding of the second extended power functor in Pstrągowski's category of synthetic spectra.

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arXiv2022

Exotic Multiplications on Periodic Complex Bordism

Victor Snaith gave a construction of periodic complex bordism by inverting the Bott element in the suspension spectrum of $BU$. This presents an $\mathbb{E}_\infty$ structure on periodic complex bordism by different means than the usual Thom spectrum definition of the $\mathbb{E}_\infty$-ring $MUP$. Here, we prove that these two $\mathbb{E}_\infty$-rings are in fact different, though the underlying $\mathbb{E}_2$-rings are equivalent. Nonetheless, we prove that both rings $\mathbb{E}_\infty$-orient $KU_2^{\wedge}$ and other forms of $K$-theory.

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arXiv2022

Odd primary analogs of Real orientations

We define, in $C_p$-equivariant homotopy theory for $p>2$, a notion of $μ_p$-orientation analogous to a $C_2$-equivariant Real orientation. The definition hinges on a $C_p$-space $\mathbb{CP}^{\infty}_{μ_p}$, which we prove to be homologically even in a sense generalizing recent $C_2$-equivariant work on conjugation spaces. We prove that the height $p-1$ Morava $E$-theory is $μ_p$-oriented and that $\mathrm{tmf}(2)$ is $μ_3$-oriented. We explain how a single equivariant map $v_1^{μ_p}:S^{2ρ} \to Σ^{\infty} \mathbb{CP}^{\infty}_{μ_p}$ completely generates the homotopy of $E_{p-1}$ and $\mathrm{tmf}(2)$, expressing a height-shifting phenomenon pervasive in equivariant chromatic homotopy theory.

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arXiv2021

Eilenberg-MacLane spectra as equivariant Thom spectra

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 $C_2$. We also establish a construction of $\mathrm{H}\underline{\mathbb{Z}}_{(p)}$, 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 $\mathrm{H}\mathbb{F}_p$ arises as the Thom spectrum of a more than double loop map.

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arXiv2020

Galois reconstruction of Artin-Tate $\mathbb{R}$-motivic spectra

We explain how to reconstruct the category of Artin-Tate $\mathbb{R}$-motivic spectra as a deformation of the purely topological $C_2$-equivariant stable category. The special fiber of this deformation is algebraic, and equivalent to an appropriate category of $C_2$-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 $\mathbb{R}$-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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arXiv2020

Equivariant nonabelian Poincar\'e duality and equivariant factorization homology of Thom spectra

In this paper, we study genuine equivariant factorization homology and its interaction with equivariant Thom spectra, which we construct using the language of parametrized higher category theory. We describe the genuine equivariant factorization homology of Thom spectra, and use this description to compute several examples of interest. A key ingredient for our computations is an equivariant nonabelian Poincar\'e duality theorem, in which we prove that factorization homology with coefficients in a $G$-space is given by a mapping space. We compute the Real topological Hochschild homology ($THR$) of the Real bordism spectrum $MU_\mathbb{R}$ and of the equivariant Eilenberg--MacLane spectra $H\underline{\mathbb{F}}_2$ and $H\underline{\mathbb{Z}}_{(2)}$, as well as factorization homology of the sphere $S^{2\sigma}$ with coefficients in these Eilenberg--MacLane spectra. In Appendix B, Jeremy Hahn and Dylan Wilson compute $THR(H\underline{\mathbb{Z}})$.

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arXiv2020

Real Orientations of Lubin-Tate Spectra

We show that Lubin-Tate spectra at the prime $2$ are Real oriented and Real Landweber exact. The proof is by 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 $E_n$ with its $C_2$-action given 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 $C_2$-fixed points.

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arXiv2019

Wilson Spaces, Snaith Constructions, and Elliptic Orientations

We construct a canonical family of even periodic $\mathbb{E}_{\infty}$-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 $\mathbb{CP}^{\infty}$. At height $2$ we replace $\mathbb{CP}^{\infty}$ with a $p$-local retract of $\mathrm{BU} \langle 6 \rangle$, 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 $\mathrm{BP}\langle n-1 \rangle$ 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 $E_n^{hS\mathbb{G}^{\pm}}$ from $\mathrm{K}(\mathbb{Z},n+1)$.

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