Towards the K(2)–local homotopy groups of Z

Bhattacharya, Prasit; Egger, Philip

doi:10.2140/agt.2020.20.1235

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Towards the

$K(2)$ –local homotopy groups of

$Z$

Prasit Bhattacharya and Philip Egger

Algebraic & Geometric Topology 20 (2020) 1235–1277

DOI: 10.2140/agt.2020.20.1235

Abstract

Previously (Adv. Math. 360 (2020) art. id. 106895), we introduced a class $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="false"><mrow><mi mathvariant="bold-script">𝒵</mi></mrow><mo class="MathClass-op">˜</mo></mover></math>$ of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn></math>$ –local finite spectra and showed that all spectra $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Z</mi> <mo class="MathClass-rel">∈</mo><mover accent="false"><mrow><mi mathvariant="bold-script">𝒵</mi></mrow><mo class="MathClass-op">˜</mo></mover></math>$ admit a $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub></math>$ –self-map of periodicity $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>1</mn></math>$ . The aim here is to compute the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>K</mi><mrow><mo class="MathClass-open">(</mo><mrow><mn>2</mn></mrow><mo class="MathClass-close">)</mo></mrow></math>$ –local homotopy groups $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>π</mi></mrow><mrow><mo class="MathClass-bin">∗</mo></mrow></msub><msub><mrow><mi>L</mi></mrow><mrow><mi>K</mi><mrow><mo class="MathClass-open">(</mo><mrow><mn>2</mn></mrow><mo class="MathClass-close">)</mo></mrow></mrow></msub><mi>Z</mi></math>$ of all spectra $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Z</mi> <mo class="MathClass-rel">∈</mo><mover accent="false"><mrow><mi mathvariant="bold-script">𝒵</mi></mrow><mo class="MathClass-op">˜</mo></mover></math>$ using a homotopy fixed point spectral sequence, and we give an almost complete answer. The incompleteness lies in the fact that we are unable to eliminate one family of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>d</mi></mrow><mrow><mn>3</mn></mrow></msub></math>$ –differentials and a few potential hidden $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn></math>$ –extensions, though we conjecture that all these differentials and hidden extensions are trivial.

Keywords

$K(2)$ –local homotopy of

$Z$ , stable homotopy,

$v_2$ –periodicity

Mathematical Subject Classification 2010

Primary: 55N20, 55Q10, 55Q51

References

Publication

Received: 17 April 2018

Revised: 22 August 2019

Accepted: 6 September 2019

Published: 27 May 2020

Authors

Prasit Bhattacharya
Department of Mathematics University of Virginia Charlottesville, VA United States

Philip Egger
Center for Neuroprosthetics Swiss Federal Institute of Technology (EPFL) Geneva Switzerland

Volume 20, issue 3 (2020)