The first k-invariant of a double loop space is trivial
type: Item
instance of: scholarly article
title: The first k-invariant of a double loop space is trivial
zbMATH Open document ID: 0672.55010
DOI: 10.1007/BF01190672
author: Dominique Arlettaz
published in: Archiv der Mathematik
publication date: 1990-01-01T00:00:00Z
review text: Let X be a connected simple CW-complex and n an integer \(\geq 2\) such that \(\pi_ iX=0\) for \(1<i<n\); the first k-invariant of X is then \(k^{n+1}[X]\in H^{n+1}(k(\pi_ 1X,1);\pi_ nX)\). The main result of this paper is the following theorem: let p be a prime number and assume that X is a connected (2p-2)-fold loop space satisfying \(\pi_ iX=0\) for \(1<i<(2p-2)\), then its first k-invariant fulfills \(L_ pk^{2p- 1}[X]=0\), where \(L_ p\) denotes the product of all primes \(q<p\). In particular when \(p=2\), the theorem asserts that the first k-invariant of a connected double loop space X is trivial: \(k^ 3[X]=0\) in \(H^ 3(K(\pi_ 1X,1);\pi_ 2X)\). Two different proofs are given: the first one is based on the study of the (co)homology suspension for Eilenberg- MacLane spaces, and the second is obtained from known results on classical cohomology operations.
reviewed by: Dominique Arlettaz
Mathematics Subject Classification ID: 55S45
Mathematics Subject Classification ID: 55P35
Mathematics Subject Classification ID: 55S10
Mathematics Subject Classification ID: 55P20
zbMATH DE Number: 4100177
zbMATH Keywords: first k-invariant
zbMATH Keywords: connected (2p-2)-fold loop space
zbMATH Keywords: first k-invariant of a connected double loop space
zbMATH Keywords: (co)homology suspension for Eilenberg- MacLane spaces
zbMATH Keywords: cohomology operations
MaRDI profile type: Publication
cites work: On the homology of the special linear group over a number field
cites work: A certain exact sequence
cites work: On the theory of obstructions
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description: scientific article; zbMATH DE number 4100177