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**On the homotopy groups of symmetric spectra.**
*(English)*
Zbl 1146.55005

There are by now several models for the stable homotopy category of spectra which have a particularly nice smash product before passage to the homotopy category. The category of symmetric spectra [M. Hovey, B. Shipley and J. Smith, J. Am. Math. Soc. 13, No. 1, 149–208 (2000; Zbl 0931.55006)] is one of them. This paper deals with some of its point set properties.

One of the peculiarities of symmetric spectra is the fact that their naively defined homotopy groups, i.e. those defined as a colimit of unstable homotopy groups, need not be isomorphic to their true homotopy groups, i.e. the groups of morphisms in the homotopy category from the spheres to the spectrum. In this paper, the author illuminates this point by studying extra algebraic structure which exists naturally on the naive homotopy groups: an action of the monoid \(M\) of injections from the set of natural numbers into itself. This action is used to gain a better understanding of many phenomena surrounding the homotopy groups of symmetric spectra. To name just two results with an immediate appeal: The author characterises the class of semistable symmetric spectra, for which it is known that the naive and the true homotopy groups agree, as those symmetric spectra for which the \(M\)-action is trivial. And he identifies the \(E^2\)-term of B. Shipley’s spectral sequence [K-Theory 19, No. 2, 155–183 (2000; Zbl 0938.55017)], which computes the true homotopy groups from the naive ones, with the Tor groups of the naive homotopy groups over the integral monoid ring of \(M\).

The exposition is both detailed and clear, and peppered with examples. This paper is certainly a must-read for those who want to use symmetric spectra as their preferred model for the stable homotopy category.

One of the peculiarities of symmetric spectra is the fact that their naively defined homotopy groups, i.e. those defined as a colimit of unstable homotopy groups, need not be isomorphic to their true homotopy groups, i.e. the groups of morphisms in the homotopy category from the spheres to the spectrum. In this paper, the author illuminates this point by studying extra algebraic structure which exists naturally on the naive homotopy groups: an action of the monoid \(M\) of injections from the set of natural numbers into itself. This action is used to gain a better understanding of many phenomena surrounding the homotopy groups of symmetric spectra. To name just two results with an immediate appeal: The author characterises the class of semistable symmetric spectra, for which it is known that the naive and the true homotopy groups agree, as those symmetric spectra for which the \(M\)-action is trivial. And he identifies the \(E^2\)-term of B. Shipley’s spectral sequence [K-Theory 19, No. 2, 155–183 (2000; Zbl 0938.55017)], which computes the true homotopy groups from the naive ones, with the Tor groups of the naive homotopy groups over the integral monoid ring of \(M\).

The exposition is both detailed and clear, and peppered with examples. This paper is certainly a must-read for those who want to use symmetric spectra as their preferred model for the stable homotopy category.

Reviewer: Markus Szymik (Bochum)

### MSC:

55P42 | Stable homotopy theory, spectra |

55U35 | Abstract and axiomatic homotopy theory in algebraic topology |

### Keywords:

symmetric spectrum### References:

[1] | M Bökstedt, Topological Hochschild homology, preprint, Bielefeld (1985) |

[2] | A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Lecture Notes in Math. 304, Springer (1972) · Zbl 0259.55004 |

[3] | M Hovey, B Shipley, J Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149 · Zbl 0931.55006 |

[4] | M Lydakis, Smash products and \(\Gamma\)-spaces, Math. Proc. Cambridge Philos. Soc. 126 (1999) 311 · Zbl 0996.55020 |

[5] | 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 |

[6] | G Segal, Categories and cohomology theories, Topology 13 (1974) 293 · Zbl 0284.55016 |

[7] | B Shipley, Symmetric spectra and topological Hochschild homology, \(K\)-Theory 19 (2000) 155 · Zbl 0938.55017 |

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