A model category structure for equivariant algebraic models

Scull, Laura

doi:10.1090/S0002-9947-07-04421-2

A model category structure for equivariant algebraic models
HTML articles powered by AMS MathViewer

by Laura Scull PDF

Trans. Amer. Math. Soc. 360 (2008), 2505-2525 Request permission

Abstract:

In the equivariant category of spaces with an action of a finite group, algebraic ‘minimal models’ exist which describe the rational homotopy for $G$-spaces which are 1-connected and of finite type. These models are diagrams of commutative differential graded algebras. In this paper we prove that a model category structure exists on this diagram category in such a way that the equivariant minimal models are cofibrant objects. We show that with this model structure, there is a Quillen equivalence between the equivariant category of rational $G$-spaces satisfying the above conditions and the algebraic category of the models.

References

A. K. Bousfield and V. K. A. M. Gugenheim, On $\textrm {PL}$ de Rham theory and rational homotopy type, Mem. Amer. Math. Soc. 8 (1976), no. 179, ix+94. MR 425956, DOI 10.1090/memo/0179
Pierre Deligne, Phillip Griffiths, John Morgan, and Dennis Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245–274. MR 382702, DOI 10.1007/BF01389853
W. G. Dwyer and J. Spaliński, Homotopy theories and model categories, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 73–126. MR 1361887, DOI 10.1016/B978-044481779-2/50003-1
M. Golasiński, Equivariant rational homotopy theory as a closed model category, J. Pure Appl. Algebra 133 (1998), no. 3, 271–287. MR 1654267, DOI 10.1016/S0022-4049(97)00127-8
P. Hirschorn, Model categories and their localizations, Amer. Math. Soc. Survey 99 (2002).
M. Hovey, Model Categories, Amer. Math. Soc. Survey 63 (1999).
J. P. May, Equivariant homotopy and cohomology theory, CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996. With contributions by M. Cole, G. Comezaña, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. MR 1413302, DOI 10.1090/cbms/091
Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432, DOI 10.1007/BFb0097438
Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295. MR 258031, DOI 10.2307/1970725
C.L. Reedy, Homotopy theory of model categories, preprint.
Laura Scull, Rational $S^1$-equivariant homotopy theory, Trans. Amer. Math. Soc. 354 (2002), no. 1, 1–45. MR 1859023, DOI 10.1090/S0002-9947-01-02790-8
Dennis Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269–331 (1978). MR 646078, DOI 10.1007/BF02684341
Georgia V. Triantafillou, Equivariant minimal models, Trans. Amer. Math. Soc. 274 (1982), no. 2, 509–532. MR 675066, DOI 10.1090/S0002-9947-1982-0675066-8