Replacing homotopy actions by topological actions. II

Author:
Larry Smith

Journal:
Trans. Amer. Math. Soc. **317** (1990), 83-90

MSC:
Primary 57S99; Secondary 55P10

MathSciNet review:
976363

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Abstract: A homotopy action of a group on a space is a homomorphism from to the group of homotopy classes of homotopy equivalences of . George Cooke developed an obstruction theory to determine if a homotopy action is equivalent up to homotopy to a topological action. The question studied in this paper is: Given a diagram of spaces with homotopy actions of and maps between them that are equivariant up to homotopy, when can the diagram be replaced by a homotopy equivalent diagram of -spaces and -equivariant maps? We find that the obstruction theory of Cooke has a natural extension to this context.

**[1]**Guy Allaud,*On the classification of fiber spaces*, Math. Z.**92**(1966), 110–125. MR**0189035****[2]**A. K. Bousfield and D. M. Kan,*Homotopy limits, completions and localizations*, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR**0365573****[3]**George Cooke,*Replacing homotopy actions by topological actions*, Trans. Amer. Math. Soc.**237**(1978), 391–406. MR**0461544**, 10.1090/S0002-9947-1978-0461544-2**[4]**W. G. Dwyer and D. M. Kan,*Function complexes for diagrams of simplicial sets*, Nederl. Akad. Wetensch. Indag. Math.**45**(1983), no. 2, 139–147. MR**705421****[5]**-,*Realizing diagrams in the homotopy category by means of diagrams of simplicial sets*.**[6]**W. G. Dwyer and D. M. Kan,*A classification theorem for diagrams of simplicial sets*, Topology**23**(1984), no. 2, 139–155. MR**744846**, 10.1016/0040-9383(84)90035-1**[7]**Edwin H. Spanier,*Algebraic topology*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210112**

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DOI:
https://doi.org/10.1090/S0002-9947-1990-0976363-3

Article copyright:
© Copyright 1990
American Mathematical Society