A class of 2-local finite spectra which admit a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math>-self-map

Bhattacharya, P. B.; Egger, Philip

doi:10.1016/j.aim.2019.106895

Cited by 9 publications

(18 citation statements)

References 17 publications

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“…The only class in higher Adams filtration which can detect β 5 η 2 is η∆κ 5 , so we conclude that M tmf (β 5 η 2 ) = η∆κ 5 [−83]. (7) We begin with M alg tmf (β 6 η 2 ) = M [26] tmf (β 6 η 2 ) = ∆ 6 ν 2 [−100]. This class does not survive in the AHSS.…”

Section: Proofmentioning

confidence: 96%

“…In [7], Bhattacharya and Egger define a class of finite spectra which admit a v 1 2 -self-map. In work in progress with Bhattacharya, we are calculating tmf -based approximations to the finite complex definition of β h i for all i ≥ 1.…”

Section: The Mahowald Invariant and Homotopymentioning

confidence: 99%

“…Inspection of the AHSS shows that there are no classes in higher Adams filtration which can detect β 5 α, so we conclude that M tmf (β 5 α) = ∆ 4 q[−84]. (7) We begin with M alg tmf (β 6 α) = M [27] tmf (β 6 α) = ∆ 6 a[−104]. There is an Adams differential d 2 (∆ 6 a) = ν∆ 6 (c 4 + ) + x 165 from which we conclude that M .…”

Section: Proofmentioning

confidence: 99%

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The motivic Mahowald invariant

Quigley

2019

Algebr. Geom. Topol.

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The 2-primary homotopy β-family, defined as the collection of Mahowald invariants of Mahowald invariants of 2 i , i ≥ 1, is an infinite collection of periodic elements in the stable homotopy groups of spheres. In this paper, we calculate tmf -based approximations to this family. Our calculations combine an analysis of the Atiyah-Hirzebruch spectral sequence for the Tate construction of tmf with trivial C 2 -action and Behrens' filtered Mahowald invariant machinery.

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Section: Proofmentioning

confidence: 96%

Section: The Mahowald Invariant and Homotopymentioning

confidence: 99%

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The motivic Mahowald invariant

Quigley

2019

Algebr. Geom. Topol.

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“…The Bailey splitting does not eliminate this potential obstruction, as Ext 2,−1+2 A(1) * (Σ 24 bo 3 , Σ 24 bo 3 ) = 0. However, by Toda's Realization Theorem [13,Section 3;39], this potential obstruction also corresponds to the existence of a different 'form' of the tmf-module tmf ∧ bo 3 , with the same homology. Since Ext s,−2+s A(2) * (bo 3 , bo 3 ) = 0 for s 3, both forms are realized.…”

Section: 4mentioning

confidence: 99%

On the ring of cooperations for 2‐primary connective topological modular forms

Behrens

Ormsby

Stapleton

et al. 2019

Journal of Topology

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We analyze the ring tmf * tmf of cooperations for the connective spectrum of topological modular forms (at the prime 2) through a variety of perspectives: (1) the E2-term of the Adams spectral sequence for tmf ∧ tmf admits a decomposition in terms of Ext groups for bo-Brown-Gitler modules, (2) the image of tmf * tmf in TMF * TMF Q admits a description in terms of 2-variable modular forms, and (3) modulo v2-torsion, tmf * tmf injects into a certain product of copies of π * TMF0(N ), for various values of N . We explain how these different perspectives are related, and leverage these relationships to give complete information on tmf * tmf in low degrees. We reprove a result of Davis-Mahowald-Rezk, that a piece of tmf ∧ tmf gives a connective cover of TMF0(3), and show that another piece gives a connective cover of TMF0(5). To help motivate our methods, we also review the existing work on bo * bo, the ring of cooperations for (2-primary) connective K-theory, and in the process give some new perspectives on this classical subject matter.for bo ∧ bo (respectively, tmf ∧ tmf) splits as a direct sum of Ext-groups for the integral (respectively, bo) Brown-Gitler spectra. Section 2.4 recalls some exact sequences used in [9] which allow for an inductive approach for computing Ext of bo-Brown-Gitler comodules, and introduces related sequences which allow for an inductive approach to Ext groups of integral Brown-Gitler comodules. Section 3This section is devoted to the motivating example of bo ∧ bo. Sections 3.1-3.3 are primarily expository, based upon the foundational work of Adams, Lellmann, Mahowald, and Milgram. We make an effort to consolidate their theorems and recast them in modern notation and terminology, and hope that this will prove a useful resource to those trying to learn the classical theory of bo-cooperations and v 1 -periodic stable homotopy. To the best of our knowledge, Sections 3.4 and 3.5 provide new perspectives on this subject. Section 3.1 is devoted to the homology of the HZ i and certain Ext A(1) * -computations relevant to the Adams spectral sequence computation of bo * bo.We shift perspectives in Section 3.2 and recall Adams's description of KU * KU in terms of numerical polynomials. This allows us to study the image of bu * bu in KU * KU as a warm-up for our study of the image of bo * bo in KO * KO.We undertake this latter study in Section 3.3, where we ultimately describe a basis of KO 0 bo in terms of the '9-Mahler basis' for 2-adic numerical polynomials with domain 2Z 2 . By studying the Adams filtration of this basis, we are able to use the above results to fully describe bo * bo mod v 1 -torsion elements.In Section 3.4, we link the above two perspectives, studying the image of bo * HZ i in KO * KO. Theorem 3.6 provides a complete description of this image (mod v 1 -torsion) in terms of the 9-Mahler basis.We conclude with Section 3.5 which studies a certain mapconstructed from Adams operations. We show that this map is an injection after applying π * and exhibit how it interacts with the Brown...

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“…To this end, we replace the

bo

‐resolution of Mahowald with the

tmf

‐based Adams spectral sequence (aka the

tmf

‐resolution),

^{tmf} E_{1}^{s, t} false(X false) = π_{t} false({tmf}^{\land s + 1} \land X false) \Rightarrow π_{t - s} X,

where

tmf

denotes the spectrum of connective topological modular forms [18]. The role that was played by Mahowald's spectrum

Y

will now be reprised by

Z

, a 2‐local finite spectrum of type 2 constructed by the third author and Egger [11], with the distinguished property that it possesses a

v_{2}^{1}

‐self map

v_{2}^{1} : {normalΣ}^{6} Z \to Z,

and that there is an equivalence

tmf \land Z ≃ k (2) .

Here

k (2)

is the height 2 connective Morava

K

‐theory spectrum.…”

Section: Introductionmentioning

confidence: 99%

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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A class of 2-local finite spectra which admit a v21-self-map

Cited by 9 publications

References 17 publications

The motivic Mahowald invariant

The motivic Mahowald invariant

On the ring of cooperations for 2‐primary connective topological modular forms

The telescope conjecture at height 2 and the tmf resolution

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