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**A universal property for \(\mathrm{Sp}(2)\) at the prime 3.**
*(English)*
Zbl 1166.55002

If \(V\) is a graded vector space then the tensor algebra \(T(V)\) is universal for \(V\) in the category of associative algebras, and in the category of associative, commutative algebras, the symmetric algebra \(S(V)\) is universal for \(V\). On the level of spaces, the Bott-Samelson theorem implies that \(H_*(\Omega \Sigma X) \cong T(\widetilde H_*(X))\), and James proved that \(\Omega \Sigma X\) is universal for a path-connected space \(X\) in the category of homotopy associative \(H\)-spaces. However, in the category of homotopy associative, homotopy commutative \(H\)-spaces, there are no universal spaces \(S(X)\) for some \(X\) such that \(H_*(S(X)) \cong S(\widetilde H_*(X))\). On the other hand, by considering the \(p\)-localized category for \(p\geq5\), several useful examples of universal spaces have been established.

In the paper under review, the authors study the \(3\)-localized case. The authors show that if \(Z\) is a \(3\)-local homotopy associative, homotopy commutative \(H\)-space with \(\pi_{13}(Z)=\pi_{17}(Z)=0\), then any map \(A \to Z\) extends to an \(H\)-map \(\text{Sp}(2)\to Z\), which is unique up to homotopy, where \(A\) is the \(7\)-skeleton of \(\text{Sp}(2)\). Moreover, it is proved that the one-to-one correspondence from the homotopy set \([A,Z]\) to the homotopy set of \(H\)-maps \(H[\text{Sp}(2),Z]\) is an isomorphism of abelian groups.

In the paper under review, the authors study the \(3\)-localized case. The authors show that if \(Z\) is a \(3\)-local homotopy associative, homotopy commutative \(H\)-space with \(\pi_{13}(Z)=\pi_{17}(Z)=0\), then any map \(A \to Z\) extends to an \(H\)-map \(\text{Sp}(2)\to Z\), which is unique up to homotopy, where \(A\) is the \(7\)-skeleton of \(\text{Sp}(2)\). Moreover, it is proved that the one-to-one correspondence from the homotopy set \([A,Z]\) to the homotopy set of \(H\)-maps \(H[\text{Sp}(2),Z]\) is an isomorphism of abelian groups.

Reviewer: Yutaka Hemmi (Kochi)