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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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