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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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