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On the groups \([X,Sp(n)]\) with \(\dim X\leq4n+2\). (English) Zbl 1170.55003
The main results that were obtained by H. Hamanaka and A. Kono [J. Math. Kyoto Univ. 43, No. 2, 333–348 (2003; Zbl 1070.55007)] and others for the group \(U(n)\) are here developed for the group \(Sp(n)\subset U(2n) \subset U(2n+1)\). For a group-like space \(G\) and a based space \(X\), the based homotopy set \([X,G]\) becomes a group by pointwise multiplication. The group under consideration is the group \(Sp_n(X) = [X,Sp(n)]\), when \(X\) is a CW-complex and \(\dim X \leq 4n+2.\) Under these conditions, a homomorphism \(\Theta_H: \widetilde{KSp}^{-2}(X) \to H^{4n+2}(X)\) is found such that the following exact sequence holds
\[ \widetilde{KSp}^{-2}(X) \to H^{4n+2}(X) \to Sp_n(X) \to \widetilde{KSp}^{-1}(X) \to 0 \]
which is natural. Moreover, the induced sequence
\[ 0 \to \operatorname{Coker} \Theta_H \to Sp_n(X) \to \widetilde{KSp}^{-1}(X) \to 0 \]
is a central extension.
These results are based on an investigation of \(\operatorname{Coker}\Theta_H\). The commutator in \(Sp_n\) is explicitly given. Some applications of the obtained results are also included.

MSC:
55Q05 Homotopy groups, general; sets of homotopy classes
55N15 Topological \(K\)-theory
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