The centralizer resolution of the 𝐾(2)-local
                    sphere at the prime 2

Henn, Hans-Werner

doi:10.1090/conm/729/14692

Cited by 7 publications

(6 citation statements)

References 22 publications

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“…We first recall some facts about the automorphism group of the supersingular elliptic curve

C

, and its associated formal group. We refer to [6, 20] for more details in this context.…”

Section: Morava Stabilizer Groups and Algebrasmentioning

confidence: 99%

“…It is observed in [6, Section 3.1; 20] that the formal group

\overset{C}{true}

and the Honda formal group

H_{2}

have isomorphic endomorphism rings. Explicitly, one gets an isomorphism

End false(\overset{C}{true} false) ≅ End false(H_{2} false)

by mapping

T \mapsto α S,

where

α = \frac{1 - 2 ω}{\sqrt{- 7}} \in W false(F_{4} false)

(for the choice of

\sqrt{- 7} \in {double-struckZ}_{2}

with

\sqrt{- 7} \equiv 1 false(\mod 4 false)

).…”

Section: Morava Stabilizer Groups and Algebrasmentioning

confidence: 99%

See 1 more Smart Citation

The telescope conjecture at height 2 and the tmf resolution

Beaudry

Behrens

Bhattacharya

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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“…We first recall some facts about the automorphism group of the supersingular elliptic curve

C

, and its associated formal group. We refer to [6, 20] for more details in this context.…”

Section: Morava Stabilizer Groups and Algebrasmentioning

confidence: 99%

“…It is observed in [6, Section 3.1; 20] that the formal group

\overset{C}{true}

and the Honda formal group

H_{2}

have isomorphic endomorphism rings. Explicitly, one gets an isomorphism

End false(\overset{C}{true} false) ≅ End false(H_{2} false)

by mapping

T \mapsto α S,

where

α = \frac{1 - 2 ω}{\sqrt{- 7}} \in W false(F_{4} false)

(for the choice of

\sqrt{- 7} \in {double-struckZ}_{2}

with

\sqrt{- 7} \equiv 1 false(\mod 4 false)

).…”

Section: Morava Stabilizer Groups and Algebrasmentioning

confidence: 99%

The telescope conjecture at height 2 and the tmf resolution

Beaudry

Behrens

Bhattacharya

et al. 2021

Journal of Topology

View full text Add to dashboard Cite

show abstract

“…Note that the art of building finite resolutions has evolved in the last fifteen years. For a long time, the role of the Galois group was not as clear as it has become recently in the work of Henn in [Hen18], so we give a revised recipe here.…”

Section: Finite Resolutions and Their Spectral Sequencesmentioning

confidence: 99%

“…In these references, the resolution is constructed for E hS 2 2 . However, using the ideas of [Hen18] it is now straightforward to construct it for E hG 2 2 . Remark 5.22.…”

Section: Topological Resolutionsmentioning

confidence: 99%