12 research outputs found

    An extension in the Adams spectral sequence in dimension 54

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    We establish a hidden extension in the Adams spectral sequence converging to the stable homotopy groups of spheres at the prime 2 in the 54-stem. This extension is exceptional in that the only proof we know proceeds via Pstragowski's category of synthetic spectra. This was the final unresolved hidden 2-extension in the Adams spectral sequence through dimension 80. We hope this provides a concise demonstration of the computational leverage provided by F2\mathbb{F}_2-synthetic spectra.Comment: 3 pages. Comments welcome

    KK-theoretic counterexamples to Ravenel's telescope conjecture

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    At each prime pp and height n+12n+1 \ge 2, we prove that the telescopic and chromatic localizations of spectra differ. Specifically, for Z\mathbb{Z} acting by Adams operations on BPn\mathrm{BP}\langle n \rangle, we prove that the T(n+1)T(n+1)-localized algebraic KK-theory of BPnhZ\mathrm{BP}\langle n \rangle^{h\mathbb{Z}} is not K(n+1)K(n+1)-local. We also show that Galois hyperdescent, A1\mathbb{A}^1-invariance, and nil-invariance fail for the K(n+1)K(n+1)-localized algebraic KK-theory of K(n)K(n)-local E\mathbb{E}_{\infty}-rings. In the case n=1n=1 and p7p \ge 7 we make complete computations of T(2)K(R)T(2)_*\mathrm{K}(R), for RR certain finite Galois extensions of the K(1)K(1)-local sphere. We show for p5p\geq 5 that the algebraic KK-theory of the K(1)K(1)-local sphere is asymptotically L2fL_2^{f}-local.Comment: 100 pages. Comments very welcom

    On the boundaries of highly connected, almost closed manifolds

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    Building on work of Stolz, we prove for integers 0d30 \le d \le 3 and k>232k>232 that the boundaries of (k1)(k-1)-connected, almost closed (2k+d)(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 HFp\mathrm{H}\mathbb{F}_p-Adams filtrations for all primes pp. We additionally prove new vanishing lines in the HFp\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 BPn\mathrm{BP} \langle n \rangle-based Adams spectral sequences.Comment: Typos corrected and exposition improved. Comments welcome

    An extension in the Adams spectral sequence in dimension 54

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    Multiplicative Structures on Moore spectra

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    In this article we show that S/8 is an E₁-algebra, S/32 is an E₂-algebra, S/ⁿ⁺¹ is an Eₙ-algebra at odd primes and, more generally, for every and there exist generalized Moore spectra of type which admit an Eₙ-algebra structure.Ph.D

    Quivers and the Adams spectral sequence

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    In this paper, we describe a novel way of identifying Adams spectral sequence E2E_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 pp-local integral homology, we are able to give a decomposition of the E2E_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 \infty-categories of the second author and Patchkoria

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

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    We explain how to reconstruct the category of Artin-Tate R\mathbb{R}-motivic spectra as a deformation of the purely topological C2C_2-equivariant stable category. The special fiber of this deformation is algebraic, and equivalent to an appropriate category of C2C_2-equivariant sheaves on the moduli stack of formal groups. As such, our results directly generalize the cofiber of τ\tau philosophy that has revolutionized classical stable homotopy theory. A key observation is that the Artin-Tate subcategory of R\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 τ\tau map, which is a feature conspicuously absent from the cellular category.Comment: 77 pages. Comments welcome
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