On the high-dimensional geography problem

Burklund, Robert; Andrew, Senger,

doi:10.48550/arxiv.2007.05127

Cited by 3 publications

(4 citation statements)

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“…It would be particularly desirable to obtain sharp results on the inertia groups of highly connected manifolds. This is because, if n ≥ 3 and n = 63, then the only remaining obstacle to a full classification of (n − 1)-connected (2n)-manifolds is the determination of certain inertia groups [BS20]. These inertia groups must be 0 in dimensions above 464, and we expect many of the lower dimensions to be accessible by careful combination of the techniques of this paper with those of [BS20].…”

Section: Introductionmentioning

confidence: 92%

“…This is because, if n ≥ 3 and n = 63, then the only remaining obstacle to a full classification of (n − 1)-connected (2n)-manifolds is the determination of certain inertia groups [BS20]. These inertia groups must be 0 in dimensions above 464, and we expect many of the lower dimensions to be accessible by careful combination of the techniques of this paper with those of [BS20]. A version of Crowley's Q-form conjecture, proved in the forthcoming PhD thesis of Nagy [Nag], should provide the geometric input necessary to reduce the study of inertia groups of highly connected manifolds entirely to problems of homotopy theory.…”

Section: Introductionmentioning

confidence: 99%

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Inertia groups in the metastable range

Burklund¹,

Hahn²,

Andrew³

2020

Preprint

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We prove that the inertia groups of all sufficiently-connected, highdimensional (2n)-manifolds are trivial. Specifically, for m 0 and k > 5/12, suppose M is a (km)-connected, smooth, closed, oriented m-manifold and Σ is an exotic m-sphere. We prove that, if M Σ is diffeomorphic to M , then Σ bounds a parallelizable manifold. Our proof is an application of higher algebra in Pstragowski's category of synthetic spectra, and builds on previous work of the authors. Contents 1. Introduction 1 2. The geometry of inertia groups 4 3. The bar spectral sequence approach to the unit of MO n 5 4. Additional background 8 5. Bounding Adams filtrations with synthetic lifts 13 6. Applications of vanishing lines 17 References 19

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Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Inertia groups in the metastable range

Burklund¹,

Hahn²,

Andrew³

2020

Preprint

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“…5 There is a significant body of work for characterizing these manifolds in the smooth category that at least goes back to Wall [Wal62], yet has recent contributions [BS20]. The most important invariant is the intersection form H u+1 (M; Z) ⊗ H u+1 (M; Z) → Z.…”

Section: Ordered Configuration Spacesmentioning

confidence: 99%

Regularity and stable ranges of FI-modules

Bahran¹

2022

Preprint

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We give refined bounds for the regularity of FI-modules and the stable ranges of FI-modules for various forms of their stabilization studied in the representation stability literature. We show that our bounds are sharp in several cases. We apply these to get explicit stable ranges for diagonal coinvariant algebras, and improve those for ordered configuration spaces of manifolds and congruence subgroups of general linear groups.

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“…This ability to "tilt" an Adams spectral sequence, first obtained for the Adams-Novikov spectral sequence using motivic methods by Gheorghe, Isaksen, Wang and Xu [41], [49], [50], has important computational consequences. For example, Burklund, Hahn and Senger use Cτ methods to prove strong results about the classical Adams spectral sequence, solving -among other things -several conjectures in geometric topology [26], [28], [29], [25].…”

Section: Derived 8-categories and Goerss-hopkins Theorymentioning

confidence: 99%

Adams spectral sequences and Franke's algebraicity conjecture

Patchkoria¹,

Pstrągowski²

2021

Preprint

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To any well-behaved homology theory we associate a derived 8-category which encodes its Adams spectral sequence. As applications, we prove a conjecture of Franke on algebraicity of certain homotopy categories and establish homotopy-coherent monoidality of the Adams filtration. 7.1. The algebraic model 87 7.2. Splittings of abelian categories and the Bousfield functor 89 7.3. Bousfield adjunction 91 7.4. The truncated thread monad 94 7.5. Proof of Franke's conjecture 99 8. Applications of the conjecture 101 8.1. Modules over ring spectra 102 8.2. Chromatic algebraicity 105 8.3. Diagram 8-categories 105 Appendix A. Comodules and quotients of abelian categories 110 References 112

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On the high-dimensional geography problem

Cited by 3 publications

References 0 publications

Inertia groups in the metastable range

Inertia groups in the metastable range

Regularity and stable ranges of FI-modules

Adams spectral sequences and Franke's algebraicity conjecture

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