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 \(X^n\).


55Q10 Stable homotopy groups
55P42 Stable homotopy theory, spectra
Full Text: DOI


[1] M G Barratt, P J Eccles, \(\Gamma^+\)-structures I: A free group functor for stable homotopy theory, Topology 13 (1974) 25 · Zbl 0292.55010
[2] A K Bousfield, E M Friedlander, Homotopy theory of \(\Gamma\)-spaces, spectra and bisimplicial sets, Lecture Notes in Mathematics 658, Springer (1978) 80 · Zbl 0405.55021
[3] A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics 304, Springer (1972) · Zbl 0259.55004
[4] A Dold, R Thom, Quasifaserungen und unendliche symmetrische Produkte, Ann. of Math. \((2)\) 67 (1958) 239 · Zbl 0091.37102
[5] W G Dwyer, J Spaliński, Homotopy theories and model categories, North-Holland (1995) 73 · Zbl 0869.55018
[6] P S Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, American Mathematical Society (2003) · Zbl 1017.55001
[7] J Lillig, A union theorem for cofibrations, Arch. Math. \((\)Basel\()\) 24 (1973) 410 · Zbl 0274.55008
[8] S Macnbsp;Lane, Categories for the working mathematician, Graduate Texts in Mathematics 5, Springer (1971) · Zbl 0232.18001
[9] M A Mandell, J P May, S Schwede, B Shipley, Model categories of diagram spectra, Proc. London Math. Soc. \((3)\) 82 (2001) 441 · Zbl 1017.55004
[10] J Milnor, On spaces having the homotopy type of a CW-complex, Trans. Amer. Math. Soc. 90 (1959) 272 · Zbl 0084.39002
[11] D Quillen, Higher algebraic \(K\)-theory. I, Lecture Notes in Mathematics 341, Springer (1973) 85 · Zbl 0292.18004
[12] C Schlichtkrull, Units of ring spectra and their traces in algebraic \(K\)-theory, Geom. Topol. 8 (2004) 645 · Zbl 1052.19001
[13] G Segal, Categories and cohomology theories, Topology 13 (1974) 293 · Zbl 0284.55016
[14] B Shipley, Symmetric spectra and topological Hochschild homology, K-Theory 19 (2000) 155 · Zbl 0938.55017
[15] N E Steenrod, A convenient category of topological spaces, Michigan Math. J. 14 (1967) 133 · Zbl 0145.43002
[16] A Strøm, The homotopy category is a homotopy category, Arch. Math. \((\)Basel\()\) 23 (1972) 435 · Zbl 0261.18015
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