The generalized Boardman homomorphisms
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.
scientific article; zbMATH DE number 2046452
scientific article