nLab equivariant rational homotopy theory

Idea

The equivariant version of rational homotopy theory, dealing with rationalization in (proper) equivariant homotopy theory, detected by Bredon equivariant rational cohomology.

Related concepts

• equivariant rationalization

• vector G-space

• equivariant chain complex, model structure on equivariant chain complexes

• equivariant dgc-algebra

• equivariant PL de Rham complex

• equivariant Sullivan model

• model structure on equivariant dgc-algebras

• Quillen adjunction between equivariant simplicial sets and equivariant connective dgc-algebras

• fundamental theorem of dg-algebraic equivariant rational homotopy theory

• rational homotopy theory

• Borel-equivariant rational homotopy theory

• rational stable homotopy theory

• rational equivariant stable homotopy theory

• rational parameterized stable homotopy theory

• equivariant differential topology

References

The original reference for finite groups is

• Georgia Triantafillou, Equivariant rational homotopy theory, chapter III of Peter 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, 1996. With contributions by M. Cole, G. Comeza˜na, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. (pdf, ISBN: 978-0-8218-0319-6)

• Georgia Triantafillou, Equivariant minimal models, Trans. Amer. Math. Soc. 274 (1982) 509-532 [jstor:1999119]

but beware that Scull 01 claims that the statement about minimal models there is not correct. Corrected statements for finite groups as well as generalization to compact Lie groups, at least to the circle group, is due to:

• Laura Scull, Rational $S^1$-equivariant homotopy theory, Transactions of the AMS, Volume 354, Number 1, Pages 1-45 2001 (pdf, doi:10.1090/S0002-9947-01-02790-8)

Further discussion in:

• Marek Golasiński, Componentwise injective models of functors to DGAs, Colloquium Mathematicum, Vol. 73, No. 1 (1997) (dml:21048, pdf)

• Marek Golasiński, Injective models of $G$-disconnected simplicial sets, Annales de l’Institut Fourier, Volume 47 (1997) no. 5, p. 1491-1522 (numdam:AIF_1997__47_5_1491_0)

The model structure on equivariant dgc-algebras, generalizing the projective model structure on dgc-algebras, in which equivariant minimal Sullivan models are cofibrant objects:

• Laura Scull, A model category structure for equivariant algebraic models, Transactions of the American Mathematical Society 360 (5), 2505-2525, 2008 (doi:10.1090/S0002-9947-07-04421-2)

partially reviewed also in

• Rekha Santhanam, Soumyadip Thandar, §2 of: Equivariant Intrinsic Formality [arXiv:2301.06824]

See also

• Peter J. Kahn, Rational Moore G-Spaces, Transactions of the American Mathematical Society Vol. 298, No. 1 (1986), pp. 245-271 (jstor:2000619)

• C. Allday, V. Puppe, sections 3.3 and 3.4 of Cohomological methods in transformation groups, Cambridge 1993 (doi:10.1017/CBO9780511526275)