The generalized Boardman homomorphisms
type: Item
instance of: scholarly article
title: The generalized Boardman homomorphisms
zbMATH Open document ID: 1047.55007
DOI: 10.2478/BF02475949
author: Dominique Arlettaz
published in: Central European Journal of Mathematics
publication date: 2004-02-23T00:00:00Z
review text: For a spectrum \(X\), the Boardman homomorphism \(b^n: \pi^n (X) = [X, S] \to H^n X = [X, \Sigma^n H {\mathbb Z}]\) is the map induced by composition with a generator: \(i: S \to H {\mathbb Z}\) of \(\pi_0 H {\mathbb Z} = {\mathbb Z}\). The author shows that if \(X\) is \(m-1\) connected and of finite dimension \(d\) then there are numbers \(\overline{\rho}_j\) such that \(\overline{\rho}_{d-n} \ker b^n =0\) for \(m \leq n \leq d-1\) and \(\overline{\rho}_{d-n-1} \text{ co} \ker b^n =0\) for \(m \leq n \leq d-2\). The numbers \(\overline{\rho}_i\) are given by \(\overline{\rho}_i = \Pi^i_{j-1} \rho_j \) where \(\rho_j\) is the exponent of \(\pi_j(S)\). Similar results are also established for the Boardman homomorphisms \(\widehat{b}_n: \pi^n(X) \to E^n(X)\), \(\widehat{b}_n: F^n(X) \to (E \wedge F)^n (X)\), \(\widetilde{b}_n: F^n(X) \to H^n (X, \pi_0 (F))\) where \(E\) and \(F\) are connective spectra. In these cases the bounds obtained involve the orders of the Postnikov invariants of \(F\). The proofs of these results are based on earlier results of the author [Comment. Math. Helv. 68, 653--672 (1993; Zbl 0968.55003), Can. J. Math. 48, 483--495 (1996; Zbl 0866.55003)] concerning orders of Postnikov invariants and exponents.
Mathematics Subject Classification ID: 55Q55
Mathematics Subject Classification ID: 55S45
Mathematics Subject Classification ID: 55N20
zbMATH DE Number: 2046452
zbMATH Keywords: Boardman homomorphism
zbMATH Keywords: cohomotopy groups
zbMATH Keywords: generalized cohomology theories
reviewed by: Keith Johnson
MaRDI profile type: Publication
cites work: Q4067782
cites work: The order of the differentials in the Atiyah-Hirzebruch spectral sequence
cites work: Exponents for extraordinary homology groups
cites work: The Exponent of the Homotopy Groups of Moore Spectra and the Stable Hurewicz Homomorphism
cites work: Q5725255
cites work: Q4061677
cites work: Poincaré duality and Postnikov factors
full work available at URL: https://doi.org/10.2478/bf02475949
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description: scientific article; zbMATH DE number 2046452