Chromatic structures in stable homotopy theory

Barthel, Tobias; Beaudry, Agnès

doi:10.1201/9781351251624-5

“…In [12], D. Clausen and A. Mathew gave a new (and short) proof of Theorem 1.1. 4, by showing that every localization of Sp, that admits a Bousfield-Kuhn functor, is 1-semiadditive. Combining this observation with the above results, the situation can be pleasantly summarized as follows:…”

Section: Resultsmentioning

confidence: 99%

Ambidexterity in chromatic homotopy theory

Carmeli

¹

,

Schlank

²

,

Yanovski

³

2022

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We extend the theory of ambidexterity developed by M. J. Hopkins and J. Lurie and show that the $$\infty $$ ∞ -categories of $$T\!\left( n\right) $$ T n -local spectra are $$\infty $$ ∞ -semiadditive for all n, where $$T\!\left( n\right) $$ T n is the telescope on a $$v_{n}$$ v n -self map of a type n spectrum. This generalizes and provides a new proof for the analogous result of Hopkins–Lurie on $$K\!\left( n\right) $$ K n -local spectra. Moreover, we show that $$K\!\left( n\right) $$ K n -local and $$T\!\left( n\right) $$ T n -local spectra are respectively, the minimal and maximal 1-semiadditive localizations of spectra with respect to a homotopy ring, and that all such localizations are in fact $$\infty $$ ∞ -semiadditive. As a consequence, we deduce that several different notions of “bounded chromatic height” for homotopy rings are equivalent, and in particular, that $$T\!\left( n\right) $$ T n -homology of $$\pi $$ π -finite spaces depends only on the nth Postnikov truncation. A key ingredient in the proof of the main result is a construction of a certain power operation for commutative ring objects in stable 1-semiadditive $$\infty $$ ∞ -categories. This is closely related to some known constructions for Morava E-theory and is of independent interest. Using this power operation we also give a new proof, and a generalization, of a nilpotence conjecture of J. P. May, which was proved by A. Mathew, N. Naumann, and J. Noel.

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“…In this paper we have not really touched on the extensive study of power operations in chromatic homotopy theory (cf. [90,7]). Given Lubin-Tate cohomology theories E and F associated to formal groups of height n at the prime p, we have both cohomology operations and power operations.…”

Section: Further Questionsmentioning

confidence: 99%

E n -spectra and Dyer-Lashof operations

Lawson¹

2020

Handbook of Homotopy Theory

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Cohomology operations are absolutely essential in making cohomology an effective tool for studying spaces. In particular, the mod-p cohomology groups of a space X are enhanced with a binary cup product, a Bockstein derivation, and Steenrod's reduced power operations; these satisfy relations such as graded-commutativity, the Cartan formula, the Adem relations, and the instability relations [92]. The combined structure of these cohomology operations is very effective in homotopy theory because of three critical properties.These operations are natural. We can exclude the possibility of certain maps between spaces because they would not respect these operations.These operations are constrained. We can exclude the existence of certain spaces because the cup product and power operations would be incompatible with the relations that must hold. AcknowledgementsThe author would like to thank for discussions related to this material, and Haynes Miller for a careful reading of an earlier draft of this paper. The author also owes a significant long-term debt to Charles Rezk for what understanding he possesses.

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“…Fix a prime

p

and let

X

be a finite spectrum. The perspective of chromatic homotopy theory is to understand

X_{false(p false)}

through the study of its chromatic tower [3; 43, Section 7.5]

\dots \to X_{E (n)} \to X_{E (n - 1)} \to \dots \to X_{E (0)} = X_{double-struckQ} .

Here,

X_{E false(n false)}

denotes the Bousfield localization of

X

with respect to the Johnson–Wilson spectrum

E (n)

with

π_{*} E false(n false) = {double-struckZ}_{(p)} false[v_{1}, \dots, v_{n}, v_{n}^{- 1} false],

where

false| v_{n} false| = 2 false(p^{n} - 1 false)

. The chromatic convergence theorem of Hopkins and Ravenel [23] states that

X_{false(p false)}

is recovered as the inverse limit of the tower.…”

Section: Introductionmentioning

confidence: 99%

“…The height

n

Morava

E

‐theory spectrum

E_{n}

[3] is a

K (n)

‐local even periodic variant of the Johnson–Wilson spectrum

E (n)

. Like

E (n)

, there are many forms of

E_{n}

, one for each height

n

formal group law

scriptF

over a perfect field

double-struckF

of characteristic

p

.…”

Section: Introductionmentioning

confidence: 99%

The telescope conjecture at height 2 and the tmf resolution

Beaudry

¹

,

Behrens

²

,

Bhattacharya

³

et al. 2021

Journal of Topology

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Mahowald proved the height 1 telescope conjecture at the prime 2 as an application of his seminal work on bo‐resolutions. In this paper, we study the height 2 telescope conjecture at the prime 2 through the lens of tmf‐resolutions. To this end, we compute the structure of the tmf‐resolution for a specific type 2 complex Z. We find that, analogous to the height 1 case, the E1‐page of the tmf‐resolution possesses a decomposition into a v2‐periodic summand, and an Eilenberg–MacLane summand which consists of bounded v2‐torsion. However, unlike the height 1 case, the E2‐page of the tmf‐resolution exhibits unbounded v2‐torsion. We compare this to the work of Mahowald–Ravenel–Shick, and discuss how the validity of the telescope conjecture is connected to the fate of this unbounded v2‐torsion: either the unbounded v2‐torsion kills itself off in the spectral sequence, and the telescope conjecture is true, or it persists to form v2‐parabolas and the telescope conjecture is false. We also study how to use the tmf‐resolution to effectively give low‐dimensional computations of the homotopy groups of Z. These computations allow us to prove a conjecture of the second author and Egger: the E(2)‐local Adams–Novikov spectral sequence for Z collapses.

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Chromatic structures in stable homotopy theory

Cited by 15 publications

References 74 publications

Ambidexterity in chromatic homotopy theory

Ambidexterity in chromatic homotopy theory

E n -spectra and Dyer-Lashof operations

The telescope conjecture at height 2 and the tmf resolution

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