An extension in the Adams spectral sequence in dimension 54

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 $\mathbb{F}_2$-synthetic spectra.Comment: 3 pages. Comments welcome

KK-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.Comment: 100 pages. Comments very welcom

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.Comment: Typos corrected and exposition improved. Comments welcome

Multiplicative Structures on Moore spectra

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

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 $\infty$-categories of the second author and Patchkoria

Galois reconstruction of Artin-Tate R\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.Comment: 77 pages. Comments welcome