A model category structure for equivariant algebraic models
Author: Laura Scull
Journal: Trans. Amer. Math. Soc. 360 (2008), 2505-2525
MSC (2000): Primary 55P91; Secondary 18G55, 55P62
Published electronically: November 28, 2007
MathSciNet review: 2373323
Abstract: In the equivariant category of spaces with an action of a finite group, algebraic `minimal models' exist which describe the rational homotopy for -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 -spaces satisfying the above conditions and the algebraic category of the models.
- [BG] A. K. Bousfield and V. K. A. M. Gugenheim, On 𝑃𝐿 de Rham theory and rational homotopy type, Mem. Amer. Math. Soc. 8 (1976), no. 179, ix+94. MR 0425956, https://doi.org/10.1090/memo/0179
- [DGMS] Pierre Deligne, Phillip Griffiths, John Morgan, and Dennis Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245–274. MR 0382702, https://doi.org/10.1007/BF01389853
- [DS] W. G. Dwyer and J. Spaliński, Homotopy theories and model categories, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 73–126. MR 1361887, https://doi.org/10.1016/B978-044481779-2/50003-1
- [G] M. Golasiński, Equivariant rational homotopy theory as a closed model category, J. Pure Appl. Algebra 133 (1998), no. 3, 271–287. MR 1654267, https://doi.org/10.1016/S0022-4049(97)00127-8
- [Hi] P. Hirschorn, Model categories and their localizations, Amer. Math. Soc. Survey 99 (2002).
- [Ho] M. Hovey, Model Categories, Amer. Math. Soc. Survey 63 (1999).
- [M] 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
- [Q1] Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432
- [Q2] Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295. MR 0258031, https://doi.org/10.2307/1970725
- [R] C.L. Reedy, Homotopy theory of model categories, preprint.
- [Sc] Laura Scull, Rational 𝑆¹-equivariant homotopy theory, Trans. Amer. Math. Soc. 354 (2002), no. 1, 1–45. MR 1859023, https://doi.org/10.1090/S0002-9947-01-02790-8
- [Su] Dennis Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269–331 (1978). MR 0646078
- [T] Georgia V. Triantafillou, Equivariant minimal models, Trans. Amer. Math. Soc. 274 (1982), no. 2, 509–532. MR 675066, https://doi.org/10.1090/S0002-9947-1982-0675066-8
- A.K. Bousfield and V.K.A.M. Gugenheim, On PL de Rham theory and rational homotopy type, Mem. Amer. Math. Soc., vol 8 no 179 (1976). MR 0425956 (54:13906)
- P. Deligne, P. Griffiths, J. Morgan and D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. vol 29 pp. 245-274 (1975). MR 0382702 (52:3584)
- W.G. Dwyer and J. Spalinski, Homotopy Theories and Model Categories, Handbook of Algebraic Topology (1995) 73-126. MR 1361887 (96h:55014)
- M. Golasinski, Equivariant rational homotopy theory as a closed model category, J. Pure and App. Algebra 133 (1998) 271-287. MR 1654267 (2000a:55025)
- 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 Lectures vol 91 (1997). MR 1413302 (97k:55016)
- D. G. Quillen, Homotopical Algebra, SLNM 43, Springer, Berlin (1967). MR 0223432 (36:6480)
- D. G. Quillen, Rational homotopy theory, Ann. Math. 90 (1969) 205-295. MR 0258031 (41:2678)
- C.L. Reedy, Homotopy theory of model categories, preprint.
- L. Scull, Rational -equivariant homotopy theory, Trans. Amer. Math. Soc., vol 354 pp. 1-45 (2001). MR 1859023 (2002g:55021)
- D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Etudes Sci. Publ. Math. vol 47 pp. 269-332 (1977). MR 0646078 (58:31119)
- G. Triantafillou, Equivariant Minimal Models, Trans. Amer. Math. Soc., vol 274 pp. 509-532 (1982). MR 675066 (84g:55017)
Affiliation: Department of Mathematics, The University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia, Canada
Keywords: Equivariant homotopy, minimal model, rational homotopy theory, model category
Received by editor(s): March 19, 2005
Received by editor(s) in revised form: February 10, 2006
Published electronically: November 28, 2007
Additional Notes: The author was supported in part by the NSERC
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.