Splitting Madsen–Tillmann spectra, II : The Steinberg idempotents and Whitehead conjecture

Kashiwabara, Takuji; Zare, Hadi

doi:10.2140/agt.2023.23.1935

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Splitting Madsen–Tillmann spectra, II: The Steinberg idempotents and Whitehead conjecture

Takuji Kashiwabara and Hadi Zare

Algebraic & Geometric Topology 23 (2023) 1935–1957

DOI: 10.2140/agt.2023.23.1935

Abstract

We show that there is a splitting of the spectrum $Σ^{- n} D (n)$ off the Madsen–Tillmann spectrum $MTO (n) = BO {(n)}^{- γ_{n}}$ compatible with the classic splitting of $M (n)$ off $BO {(n)}_{+}$ , localized at the prime $p = 2$ . For $n = 2$ , together with our previous splitting result on Madsen–Tillmann spectra, this shows that $MTO (2)$ is homotopy equivalent to $BSO {(3)}_{+} \lor Σ^{- 2} D (2)$ . We also discuss its implication for characteristic classes.

Dedicated to the memory of Stephen A Mitchell

Keywords

Madsen–Tillmann spectra, Whitehead conjecture, Steinberg idempotent

Mathematical Subject Classification 2010

Primary: 55P42, 55P47, 55R40, 57R20

Secondary: 55R35, 55S12, 55S15, 57N70

References

Publication

Received: 8 December 2015

Revised: 20 December 2021

Accepted: 6 March 2022

Published: 25 July 2023

Authors

Takuji Kashiwabara
Laboratoire de Mathématiques Institut Fourier, Université Grenoble Alpes Grenoble France

Hadi Zare
School of Mathematics, Statistics and Computer Science College of Science University of Tehran Tehran Iran	School of Mathematics Institute for Research in Fundamental Sciences (IPM) Tehran Iran

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Volume 23, issue 5 (2023)