Inertia groups in the metastable range

Burklund, Robert; Hahn, Jeremy; Andrew, Senger,

doi:10.48550/arxiv.2010.09869

Cited by 3 publications

(3 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The problem of determining the inertia groups is somewhat subtle, but tractable [90,Theorem D]. In later work which builds on the methods developed in this paper the authors have analyzed boundary spheres and inertia groups substantially deeper into the metastable range [23].…”

Section: The Classification Of (N−1)-connected 2n-manifoldsmentioning

confidence: 99%

On the boundaries of highly connected, almost closed manifolds

Burklund,

Hahn,

Senger

2023

Acta Mathematica

View full text Add to dashboard Cite

Theorem 1.4. (Conjecture of Galatius and Randal-Williams) Let MO⟨4n⟩ denote the Thom spectrum of the canonical map τ ⩾4n BO −! BO, where τ ⩾4n BO denotes the (4n−1)-connected cover of BO. For all n>31, the unit mapis surjective, with kernel exactly the image of the J-homomorphism π 8n−1 O −! π 8n−1 S .We label Theorem 1.4 a conjecture of Galatius and Randal-Williams since it is, when n>31, equivalent to Conjectures A and B of their work [35]. Theorem 1.4 allows us to improve the bound k>232 in the d=0 case of Theorem 1.1. For details, see Theorem 8.6.Remark 1.5. Much of Theorem 1.4 is classical: the surjectivity statement follows from surgery as in [50, Theorem 6.6], while the Pontryagin-Thom correspondence guarantees that the image of the J-homomorphism is contained in the kernel of the unit map π 8n−1 S!π 8n−1 MO⟨4n⟩. The difficult point is to prove that the kernel of this unit map contains only the image of J.A priori, there could be additional elements in this kernel, and the concern has a geometric interpretation. Let Σ Q ∈Θ 8n−1 denote the boundary of the manifold obtained by plumbing together two copies of the 4n-dimensional linear disk bundle over S 4n that generates the image of π 4n BSO(4n−1) in π 4n BSO(4n). Theorem 1.4 is equivalent to the claim that, for n>31, the class [Σ Q ]∈coker(J) 8n−1 is trivial [90, Lemma 10.3].Our proof of Theorem 1.4 follows a general strategy due to Stolz [90], which he applied to prove some cases of Theorem 1.1. For each prime number p we compute a lower bound on the HF p -Adams filtrations of classes in the kernel of the unit mapOur lower bound is given in Theorem 10.8, and it is one of the main technical achievements of this paper. It is approximately double the bound obtained by Stolz in [90, Satz 12.7], and we devote § §4-6 and § §9-10 to its proof.Remark 1.6. A key portion of the argument for Theorem 10.8 takes place in Pstrągowski's category of synthetic spectra [77] (cf. [37] for an alternative construction of BP-synthetic spectra). Other users of this category may be interested in our omnibus Theorem 9.19, which relates Adams spectral sequences to synthetic homotopy groups.r. burklund, j. hahn and a. senger for the spectrum Y =C(2)⊗C(η). In more classical language this result is known to experts, and follows from combining Miller's tools with computational results of Davis and Mahowald [28]. In §15, we establish a v 1 -banded vanishing line in the modified HF 2 -Adams spectral sequence for the Moore spectrum C(8) and conclude, in particular, Theorem 1.9.Appendix A. The first part of this appendix is devoted to a technical proof of Theorem 9.19. The theorem provides the means to translate statements about E-based Adams spectral sequences into statements about E-based synthetic spectra, and vice-versa. The proofs in this section are mostly a matter of careful book-keeping.The second part of the appendix contains a computation of the HF 2 -synthetic homotopy groups of the 2-complete sphere through the Toda range. We find that this computation illustrates many of the s...

show abstract

Section: The Classification Of (N−1)-connected 2n-manifoldsmentioning

confidence: 99%

On the boundaries of highly connected, almost closed manifolds

Burklund,

Hahn,

Senger

2023

Acta Mathematica

View full text Add to dashboard Cite

show abstract

“…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

View full text Add to dashboard Cite

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

show abstract

“…On the other hand, [Pst18] has proposed another explanation of the τ -deformation picture, by introducing the algebraic notion of synthetic spectra. These have since found other applications, for instance to asymptotic chromatic algebraicity [Pst21], Goerss-Hopkins obstruction theory [PV21], and questions of manifold geometry in [BHS19], [BHS20a], and have been expanded in scope in [PP21]. Using the same filtered module spectra technique, which we use in the p-complete setting to prove Theorem 3, we obtain in Subsection 1.7 an integral comparison with synthetic spectra.…”

Section: Introductionmentioning

confidence: 99%

Moduli stack of oriented formal groups and cellular motivic spectra over $\mathbf C$

Gregoric¹

2021

Preprint

View full text Add to dashboard Cite

We exhibit a relationship between motivic homotopy theory and spectral algebraic geometry, based on the motivic τ -deformation picture of Gheorghe, Isaksen, Wang, Xu. More precisely, we identify cellular motivic spectra over C with ind-coherent sheaves (in a slighly non-standard sense) on a certain spectral stack τ≥0(M or FG ). The latter is the connective cover of the non-connective spectral stack M or FG , the moduli stack of oriented formal groups, which we have introduced previously and studied in connection with chromatic homotopy theory. We also provide a geometric origin on the level of stacks for the observed τ -deformation behavior on the level of sheaves, based on a notion of extended effective Cartier divisors in spectral algebraic geometry.

show abstract

Inertia groups in the metastable range

Cited by 3 publications

References 25 publications

On the boundaries of highly connected, almost closed manifolds

On the boundaries of highly connected, almost closed manifolds

Adams spectral sequences and Franke's algebraicity conjecture

Moduli stack of oriented formal groups and cellular motivic spectra over $\mathbf C$

Contact Info

Product

Resources

About