The Kervaire invariant of immersions
type: Item
instance of: scholarly article
title: The Kervaire invariant of immersions
zbMATH Open document ID: 0576.57032
DOI: 10.1007/BF01388658
published in: Inventiones Mathematicae
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143187
review text: The authors study E. H. Brown's generalized Kervaire invariant \(K: \Omega\) \({}_*^ G\to {\mathbb{Z}}/8\) where \(\Omega_*^ G\) is an appropriate cobordism theory of immersed submanifolds in Euclidean space. Let \(G=O(1)\) or SO(2). The main results are: (1) unless \(n=1\) or 3 and \(G=O(1)\), the Kervaire invariant of a class \(x\in \Omega^ G_{2n}\) is nonzero if and only if x is represented in the Adams spectral sequence by a class in \(Ext_ A(H^*MG,{\mathbb{Z}}/2)\) in the image of multiplication by \(h^ 2_ j\) and (2) let \(u_*: Ext_ A({\mathbb{Z}}/2,{\mathbb{Z}}/2)\to Ext_ A(H^*MSO(2),{\mathbb{Z}}/2)\) be induced by the Thom class \(u: S^ 0\to MSO(2)\). Then for every \(j\geq 1\), there exists an element \(\theta_ jSO(2)\) represented by \(u_*(h^ 2_ j)\); thus \(\theta_ jS0(2)\) has nonzero Kervaire invariant in this theory. The result given by (1) generalizes results of \textit{W. Browder} [Ann. Math., II. Ser. 90, 157-186 (1969; Zbl 0198.285)]
Mathematics Subject Classification ID: 57R90
Mathematics Subject Classification ID: 57R40
Mathematics Subject Classification ID: 55Q45
Mathematics Subject Classification ID: 55Q10
Mathematics Subject Classification ID: 55N91
zbMATH DE Number: 3921558
zbMATH Keywords: generalized Kervaire invariant
zbMATH Keywords: cobordism theory of immersed submanifolds
zbMATH Keywords: Adams spectral sequence
author: Ralph L. Cohen
author: John D. S. Jones
author: Mark Mahowald
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3921558