## The homotopy infinite symmetric product represents stable homotopy.(English)Zbl 1131.55007

The infinite symmetric product $$SP(X)$$ of a based $$CW$$-complex $$X$$ is defined by a colimit of a diagram. The author introduces $$SP_h(X)$$ defined by a homotopy colimit of the diagram, and shows the homotopy equivalence $$SP_h^\wedge (X)=\Omega B(SP_h(X))\simeq Q(X)=$$ hocolim$$_n\Omega^n\Sigma^nX$$.
In particular, if $$X$$ is connected, then $$\pi_*(SP(X))=\widetilde{H}_*(X)$$ and $$SP_h(X)=SP_h^\wedge(X)$$, and the canonical map $$SP_h(X)\to SP(X)$$ induces the Hurewicz homomorphism $$\pi_*^S(X)\to \widetilde{H}_*(X)$$. The homotopy equivalence implies the theorem of Barratt-Priddy-Quillen relating $$Q(X)$$ to the action of the symmetric groups on the spaces $$X^n$$.

### MSC:

 55Q10 Stable homotopy groups 55P42 Stable homotopy theory, spectra
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### References:

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