Synthetic spectra and the cellular motivic category

Pstrągowski, Piotr

doi:10.1007/s00222-022-01173-2

Cited by 7 publications

(21 citation statements)

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“…In fact, the class of epimorphisms is enough to describe canonically the abelian category whose Ext-groups form the E 2 -page. To do so, we consider sheaves with respect to a certain topology, as in the construction of synthetic spectra [Pst22].…”

Section: Milnor Modulesmentioning

confidence: 99%

“…Remark 4.9. A reader familiar with [Pst22] might recall that in the construction of synthetic spectra, one works with the site of those finite spectra X, such that K * X is projective as a K * -module [Pst22, Definition 3.12]. Since Morava K -theories are fields, this yields the class of all finite spectra, as above.…”

Section: Milnor Modulesmentioning

confidence: 99%

“…Remark 4.11. A reader familiar with synthetic spectra might immediately recognise that the category of Milnor modules can be canonically identified as Mil abs Syn ♥ K , the heart of the stable ∞-category of K -based synthetic spectra of [Pst22].…”

Section: Milnor Modulesmentioning

confidence: 99%

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MORAVA K-THEORY AND FILTRATIONS BY POWERS

Barthel¹,

Pstrągowski

2023

J. Inst. Math. Jussieu

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We prove the convergence of the Adams spectral sequence based on Morava K-theory and relate it to the filtration by powers of the maximal ideal in the Lubin–Tate ring through a Miller square. We use the filtration by powers to construct a spectral sequence relating the homology of the K-local sphere to derived functors of completion and express the latter as cohomology of the Morava stabiliser group. As an application, we compute the zeroth limit at all primes and heights.

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Section: Milnor Modulesmentioning

confidence: 99%

Section: Milnor Modulesmentioning

confidence: 99%

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MORAVA K-THEORY AND FILTRATIONS BY POWERS

Barthel¹,

Pstrągowski

2023

J. Inst. Math. Jussieu

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“…Synthetic spectra were developed by Piotr Pstrągowski in [77]. They constitute a homotopy theory, or symmetric monoidal stable ∞-category, of formal Adams spectral sequences.…”

Section: An Outline Of the Papermentioning

confidence: 99%

“…We believe that questions involving the Adams filtrations of homotopies are greatly clarified by recent work of Piotr Pstrągowski, and in particular his development of the category of synthetic spectra [77]. For E an Adams-type homology theory, E -based synthetic spectra form an ∞-category Syn E of formal E -based Adams spectral sequences.…”

Section: Synthetic Spectramentioning

confidence: 99%

On the boundaries of highly connected, almost closed manifolds

Burklund,

Hahn,

Senger

2023

Acta Mathematica

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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...

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The Adams differentials on the classes $h_{j}^{3}$

Burklund,

2024

Invent. math.

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In filtration 1 of the Adams spectral sequence, using secondary cohomology operations, Adams (Ann. Math. (2) 72:20–104 1960) computed the differentials on the classes $h_{j}$ h j , resolving the Hopf invariant one problem. In Adams filtration 2, using equivariant and chromatic homotopy theory, Hill–Hopkins–Ravenel (Ann. Math. (2) 184(1):1–262 2016) proved that the classes $h_{j}^{2}$ h j 2 support non-trivial differentials for $j \geq 7$ j ≥ 7 , resolving the celebrated Kervaire invariant one problem. The precise differentials on the classes $h_{j}^{2}$ h j 2 for $j \geq 7$ j ≥ 7 and the fate of $h_{6}^{2}$ h 6 2 remains unknown. In this paper, in Adams filtration 3, we prove an infinite family of non-trivial $d_{4}$ d 4 -differentials on the classes $h_{j}^{3}$ h j 3 for $j \geq 6$ j ≥ 6 , confirming a conjecture of Mahowald. Our proof uses two different deformations of stable homotopy theory—ℂ-motivic stable homotopy theory and $\mathbb{F}_{2}$ F 2 -synthetic homotopy theory—both in an essential way. Along the way, we also show that $h_{j}^{2}$ h j 2 survives to the Adams $E_{5}$ E 5 -page and that $h_{6}^{2}$ h 6 2 survives to the Adams $E_{9}$ E 9 -page.

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Synthetic spectra and the cellular motivic category

Cited by 7 publications

References 30 publications

MORAVA K-THEORY AND FILTRATIONS BY POWERS

MORAVA K-THEORY AND FILTRATIONS BY POWERS

On the boundaries of highly connected, almost closed manifolds

The Adams differentials on the classes $h_{j}^{3}$

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