Building on work of Stolz, we prove for integers 0 ≤ d ≤ 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 HFp-Adams filtrations for all primes p. We additionally prove new vanishing lines in the HFp-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 BP n -based Adams spectral sequences.

We prove that the local A 1 -degree of a polynomial function at an isolated zero with finite separable residue field is given by the trace of the local A 1 -degree over the residue field. This fact was originally suggested by Morel's work on motivic transfers, and by Kass and Wickelgren's work on the Scheja-Storch bilinear form. As a corollary, we generalize a result of Kass and Wickelgren relating the Scheja-Storch form and the local A 1 -degree.

We explain how to reconstruct the category of Artin-Tate 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 τ 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 τ map, which is a feature conspicuously absent from the cellular category. Contents 1. Introduction 1 2. Constructing the element ta 17 3. The ta-local category 19 4. Galois reconstruction 24 5. Modules over the cofiber of ta 30 6. The Chow t-structure and ν R 44 7. The a-local category 53 8. Completions 55 9. Odd primes 60 10. Examples and computations 61 Appendix A. Recollections on compact rigid generation 65 Appendix B. Recollections on filtered objects 68 Appendix C. A machine for deforming homotopy theories 70 References 75

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-synthetic spectra.

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