Equivariant maps which are self homotopy equivalences

Dror, E.; Dwyer, W. G.; Kan, D. M.

doi:10.1090/S0002-9939-1980-0587952-1

Equivariant maps which are self homotopy equivalences
HTML articles powered by AMS MathViewer

by E. Dror, W. G. Dwyer and D. M. Kan PDF

Proc. Amer. Math. Soc. 80 (1980), 670-672 Request permission

Abstract:

The aim of this note is (i) to give (in §2) a precise statement and proof of the (to some extent well-known) fact that the most elementary homotopy theory of “simplicial sets on which a fixed simplicial group H acts” is equivalent to the homotopy theory of “simplicial sets over the classifying complex $\bar WH$", and (ii) to use this (in §1) to prove a classification theorem for simplicial sets with an H-action, which provides classifying complexes for their equivariant maps which are self homotopy equivalences.

References

E. Dror, W. G. Dwyer, and D. M. Kan, Automorphisms of fibrations, Proc. Amer. Math. Soc. 80 (1980), no. 3, 491–494. MR 581012, DOI 10.1090/S0002-9939-1980-0581012-1
W. G. Dwyer and D. M. Kan, Calculating simplicial localizations, J. Pure Appl. Algebra 18 (1980), no. 1, 17–35. MR 578563, DOI 10.1016/0022-4049(80)90113-9

Function complexes in homotopical algebra

J. Peter May, Simplicial objects in algebraic topology, Van Nostrand Mathematical Studies, No. 11, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0222892
Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432