arXiv subjects

Andrew Senger

Publications and source records attributed to Andrew Senger.

14 recordsLinked to original sources

arXiv2025

Splitting the Madsen-Tillmann Spectra $MT\theta_n$

We prove that the Madsen-Tillmann spectrum $MT\theta_n$ splits into the sum of spectra $\Sigma^{-2n}MO\langle n+1 \rangle \oplus \Sigma^{\infty-2n}\mathbb{R} P^\infty_{2n}$ after Postnikov trunctation $\tau_{\leq \ell}$ for $\ell = \lfloor \frac{n}{2} \rfloor - 6$. To accomplish this, we prove that the connecting map in a certain fiber sequence is nullhomotopic in this range by an Adams filtration argument. As an application, we compute $H_2(B\operatorname{Diff}(W^{2n}_{g},D^{2n});\mathbb{Z})$ up to extensions for $n \geq 16$ and $g \geq 7$.

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arXiv2024

Exact bounds for even vanishing of $K_* (\mathbb{Z}/p^n)$

In this note, we prove that $K_{2i} (\mathbb{Z}/p^n) \neq 0$ if and only if $p-1$ divides $i$ and $0 \leq i \leq (p-1) p^{n-2}$, refining the even vanishing theorem of Antieau, Nikolaus and the first author in this case. As a corollary of our proof, we determine that the nilpotence order of $v_1$ in $\pi_* K(\mathbb{Z}/p^n)/p$ is equal to $\frac{p^n-1}{p-1}$. Our proof combines the recent crystallinity result for reduced syntomic cohomology of Hahn, Levy and the second author with the explicit complex computing the syntomic cohomology of $\mathcal{O}_K /\varpi^n$ constructed by Antieau, Nikolaus and the first author.

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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

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

Inertia groups of $(n-1)$-connected $2n$-manifolds

In this paper, we compute the inertia groups of $(n-1)$-connected, smooth, closed, oriented $2n$-manifolds where $n \geq 3$. As a consequence, we complete the diffeomorphism classification of such manifolds, finishing a program initiated by Wall sixty years ago, with the exception of the $126$-dimensional case of the Kervaire invariant one problem. In particular, we find that the inertia group always vanishes for $n \neq 4,8,9$ -- for $n \gg 0$, this was known by the work of several previous authors, including Wall, Stolz, and Burklund and Hahn with the first named author. When $n = 4,8,9$, we apply Kreck's modified surgery and a special case of Crowley's $Q$-form conjecture, proven by Nagy, to compute the inertia groups of these manifolds. In the cases $n=4,8$, our results recover unpublished work of Crowley--Nagy and Crowley--Olbermann. In contrast, we show that the homotopy and concordance inertia groups of $(n-1)$-connected, smooth, closed, oriented $2n$-manifolds with $n \geq 3$ always vanish.

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arXiv2022

The Brown-Peterson spectrum is not $\mathbb{E}_{2(p^2+2)}$ at odd primes

We show that the odd-primary Brown-Peterson spectrum $\mathrm{BP}$ does not admit the structure of an $\mathbb{E}_{2(p^2+2)}$ ring spectrum and that there can be no map $\mathrm{MU} \to \mathrm{BP}$ of $\mathbb{E}_{2p+3}$ ring spectra. We also prove the same results for truncated Brown-Peterson spectra $\mathrm{BP} \langle n \rangle$ of height $n \geq 4$. This extends results of Lawson at the prime $2$.

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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

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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arXiv2022

Obstruction theory and the level $n$ elliptic genus

Given a height $\leq 2$ Landweber exact $\mathbb{E}_\infty$-ring $E$ whose homotopy is concentrated in even degrees, we show that any complex orientation of $E$ which satisfies the Ando criterion admits a unique lift to an $\mathbb{E}_\infty$-complex orientation $\mathrm{MU} \to E$. As a consequence, we give a short proof that the level $n$ elliptic genus lifts uniquely to an $\mathbb{E}_\infty$-complex orientation $\mathrm{MU} \to \mathrm{tmf}_1 (n)$ for all $n \geq 2$.

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arXiv2022

How Big are the Stable Homotopy Groups of Spheres?

In this article we show that the $p$-torsion exponent of the stable stems grows sublinearly in $n$ and the $p$-rank of the $E_2$-page of the Adams spectral sequence grows as $\exp(Θ( \log(n)^3))$. 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 $E_2$-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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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

On the high-dimensional geography problem

In 1962, Wall showed that smooth, closed, oriented, $(n-1)$-connected $2n$-manifolds of dimension at least $6$ are classified up to connected sum with an exotic sphere by an algebraic refinement of the intersection form which he called an $n$-space. In this paper, we complete the determination of which $n$-spaces are realizable by smooth, closed, oriented, $(n-1)$-connected $2n$-manifolds for all $n \neq 63$. In dimension $126$ the Kervaire invariant one problem remains open. Along the way, we completely resolve conjectures of Galatius-Randal-Williams and Bowden-Crowley-Stipsicz, showing that they are true outside of the exceptional dimension $23$, where we provide a counterexample. This counterexample is related to the Witten genus and its refinement to a map of $\mathbb{E}_\infty$-ring spectra by Ando-Hopkins-Rezk. By previous work of many authors, including Wall, Schultz, Stolz and Hill-Hopkins-Ravenel, as well as recent joint work of Hahn with the authors, these questions have been resolved for all but finitely many dimensions, and the contribution of this paper is to fill in these gaps.

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arXiv2014

Partition Statistics Equidistributed with the Number of Hook Difference One Cells

Let $λ$ be a partition, viewed as a Young diagram. We define the hook difference of a cell of $λ$ to be the difference of its leg and arm lengths. Define $h_{1,1}(λ)$ to be the number of cells of $λ$ with hook difference one. In the paper of Buryak and Feigin (arXiv:1206.5640), algebraic geometry is used to prove a generating function identity which implies that $h_{1,1}$ is equidistributed with $a_2$, the largest part of a partition that appears at least twice, over the partitions of a given size. In this paper, we propose a refinement of the theorem of Buryak and Feigin and prove some partial results using combinatorial methods. We also obtain a new formula for the q-Catalan numbers which naturally leads us to define a new q,t-Catalan number with a simple combinatorial interpretation.

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