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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
Abstract

We show that there is a splitting of the spectrum ΣnD(n) off the Madsen–Tillmann spectrum MTO (n) = BO (n)γn 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)+ Σ2D(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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