## 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.

### MSC:

 55P45 $$H$$-spaces and duals 55Q15 Whitehead products and generalizations

### Keywords:

$$H$$-space; $$H$$-map; Whitehead product
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