Fixed set systems of equivariant infinite loop spaces

Authors:
Steven R. Costenoble and Stefan Waner

Journal:
Trans. Amer. Math. Soc. **326** (1991), 485-505

MSC:
Primary 55P91; Secondary 55N91, 55P47, 55R35

DOI:
https://doi.org/10.1090/S0002-9947-1991-1012523-4

MathSciNet review:
1012523

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Abstract: We develop machinery enabling us to show that suitable -spaces, including the equivariant version of , are equivariant infinite loop spaces. This involves a "recognition principle" for systems of spaces which behave formally like the system of fixed sets of a -space; that is, we give a necessary and sufficient condition for such a system to be equivalent to the fixed set system of an equivariant infinite loop space. The advantage of using the language of fixed set systems is that one can frequently replace the system of fixed sets of an actual -space by an equivalent formal system which is considerably simpler, and which admits the requisite geometry necessary for delooping. We also apply this machinery to construct equivariant Eilenberg-Mac Lane spaces corresponding to Mackey functors.

**[CW]**J. Caruso and S. Waner,*An approximation theorem for equivariant loop spaces in the compact Lie case*, Pacific J. Math.**117**(1985), no. 1, 27–49. MR**777436****[CHMW]**S. R. Costenoble, H. Hauschild, J. P. May, and S. Waner,*Equivariant infinite loop space theory*(to appear).**[D1]**Andreas W. M. Dress,*Contributions to the theory of induced representations*, Algebraic 𝐾-theory, II: “Classical” algebraic 𝐾-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Springer, Berlin, 1973, pp. 183–240. Lecture Notes in Math., Vol. 342. MR**0384917****[D2]**Andreas W. M. Dress,*Induction and structure theorems for orthogonal representations of finite groups*, Ann. of Math. (2)**102**(1975), no. 2, 291–325. MR**0387392**, https://doi.org/10.2307/1971033**[E]**A. D. Elmendorf,*Systems of fixed point sets*, Trans. Amer. Math. Soc.**277**(1983), no. 1, 275–284. MR**690052**, https://doi.org/10.1090/S0002-9947-1983-0690052-0**[HW]**Henning Hauschild and Stefan Waner,*The equivariant Dold theorem mod 𝑘 and the Adams conjecture*, Illinois J. Math.**27**(1983), no. 1, 52–66. MR**684540****[LMM]**G. Lewis, J. P. May, and J. McClure,*Ordinary 𝑅𝑂(𝐺)-graded cohomology*, Bull. Amer. Math. Soc. (N.S.)**4**(1981), no. 2, 208–212. MR**598689**, https://doi.org/10.1090/S0273-0979-1981-14886-2**[LMS]**L. G. Lewis Jr., J. P. May, M. Steinberger, and J. E. McClure,*Equivariant stable homotopy theory*, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, Berlin, 1986. With contributions by J. E. McClure. MR**866482****[M1]**J. P. May,*The geometry of iterated loop spaces*, Springer-Verlag, Berlin-New York, 1972. Lectures Notes in Mathematics, Vol. 271. MR**0420610****[M2]**-,*Classifying spaces and fibrations*, Mem. Amer. Math. Soc., no. 155 (1972).**[M3]**J. Peter May,*𝐸_{∞} ring spaces and 𝐸_{∞} ring spectra*, Lecture Notes in Mathematics, Vol. 577, Springer-Verlag, Berlin-New York, 1977. With contributions by Frank Quinn, Nigel Ray, and Jørgen Tornehave. MR**0494077****[S]**Graeme Segal,*Configuration-spaces and iterated loop-spaces*, Invent. Math.**21**(1973), 213–221. MR**0331377**, https://doi.org/10.1007/BF01390197**[W1]**Stefan Waner,*Equivariant homotopy theory and Milnor’s theorem*, Trans. Amer. Math. Soc.**258**(1980), no. 2, 351–368. MR**558178**, https://doi.org/10.1090/S0002-9947-1980-0558178-7**[W2]**Stefan Waner,*Equivariant classifying spaces and fibrations*, Trans. Amer. Math. Soc.**258**(1980), no. 2, 385–405. MR**558180**, https://doi.org/10.1090/S0002-9947-1980-0558180-5**[W3]**-,*Three topological categories of*-*spaces*, Hofstra University (preprint).

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DOI:
https://doi.org/10.1090/S0002-9947-1991-1012523-4

Article copyright:
© Copyright 1991
American Mathematical Society