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An algebraic model for \(G\)-simple homotopy types. (English) Zbl 0564.55010

The authors’ aim is to construct algebraic models for G-complexes X (G finite acting on X with nonvoid 1-connected fixed point sets \(X^ H\), for any subgroup H) which are adequate for the description of G-(simple) homotopy types and then to obtain classification results for G-(simple) homotopy types up to finite ambiguity, along the lines indicated by D. Sullivan [Publ. Math., Inst. Hautes Étud. Sci. 47 (1977), 269-331 (1978; Zbl 0374.57002)] for ordinary homotopy types. The generalization is twofold: first, in the passage to the equivariant case, they obtain G- versions of results due to D. Quillen [Ann. Math., II. Ser. 90, 205-295 (1969; Zbl 0191.537)] and H. J. Baues and J. M. Lemaire [Math. Ann. 225, 219-245 (1977; Zbl 0322.55019)] and establish the finiteness of the G-homotopy types with given G-DG Lie minimal model, integral structure and torsion bound. The same result was previously obtained by the second author [Astérisque 113/114, 312-337 (1984; see the previous review)] in terms of G-DG algebra models. The second point (motivating the Lie approach) is the construction of a G-DG Lie simple model for finite G-complexes, which is used in the place of the minimal Lie model in order to obtain the corresponding finiteness result for G- simple homotopy types. Finiteness results for G-(simple) homotopy types of \(Z_{p^ k}\) actions on spheres and of semilinear finite group actions are derived.
Reviewer: S.Papadima

MSC:

55P91 Equivariant homotopy theory in algebraic topology
57S17 Finite transformation groups
55P62 Rational homotopy theory
57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
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References:

[1] Adams, J.F., Hilton, P.J.: On the chain algebra of a loop space. Comment. Math. Helv.30, 305-330 (1956) · Zbl 0071.16403 · doi:10.1007/BF02564350
[2] Bass, H.: Algebraic K-theory. New York: Benjamin 1968 · Zbl 0174.30302
[3] Baues, H.J., Lemaire, J.M.: Minimal models in homotopy theory. Math. Ann.225, 219-242 (1977) · Zbl 0332.55013 · doi:10.1007/BF01425239
[4] Borel, A., Harish-Chandra: Arithmetic subgroups of algebraic groups. Ann. Math.75, 485-535 (1962) · Zbl 0107.14804 · doi:10.2307/1970210
[5] Bousfield, A.K., Kan, D.M.: Homotopy limits, completions, and localizations. Lecture Notes in Math. 304. Berlin, Heidelberg, New York: Springer 1972 · Zbl 0259.55004
[6] Bredon, G.E.: Equivariant cohomology theories. Lecture Notes in Math. 34. Berlin, Heidelberg, New York: Springer-Verlag 1967 · Zbl 0162.27202
[7] Tom Dieck, T., Petrie, T.: The homotopy structure of finite group actions on spheres. Lecture Notes in Math. 741, pp. 222-243. Berlin, Heidelberg, New York: Springer 1979 · Zbl 0416.57019
[8] Dovermann, K.H., Rothenberg, M.: An equivariant surgery sequence and diffeomorphism and homeomorphism classification. Preprint, 1983 · Zbl 0433.57018
[9] Elmendorf, A.D.: Systems of fixed point sets. Trans. Am. Math. Soc.277, 275-284 (1983) · Zbl 0521.57027 · doi:10.1090/S0002-9947-1983-0690052-0
[10] Illman, S.: Whitehead torsion and group action. Ann. Acad. Sci. Fenn. Ser. AI Math. No. 588 (1974) · Zbl 0303.57006
[11] May, J.P.: Simplicial objects in algebraic topology. Van Nostrand, Princeton, NJ, 1967 · Zbl 0165.26004
[12] Milnor, J.W.: Whitehead torsion. Bull. Amer. Math. Soc.72, 358-426 (1966) · Zbl 0147.23104 · doi:10.1090/S0002-9904-1966-11484-2
[13] Neisendorfer, J.: Lie algebras, coalgebras and rational homotopy theory for nilpotent spaces. Pac. J. Math.74, 429-460 (1978) · Zbl 0386.55016
[14] Quillen, D.: Rational homotopy theory. Ann. Math.90, 205-295 (1969) · Zbl 0191.53702 · doi:10.2307/1970725
[15] Rothenberg, M.: Torsion invariants and finite transformation groups. Proc. Sympos. Pure Math. AMS32, 267-313 (1978) · Zbl 0426.57013
[16] Rothenberg, M.: Homotopy type ofG-spheres. Lecture Notes in Math. 763, pp. 573-590. Berlin, Heidelberg, New York: Springer, 1979
[17] Sullivan, D.: Infinitesimal computations in topology. Publ. Math. IHES No.47, 269-331 (1978) · Zbl 0374.57002
[18] Triantafillou, G.: Equivariant minimal models. Trans. Amer. Math. Soc.274, 509-532 (1982) · Zbl 0516.55010 · doi:10.1090/S0002-9947-1982-0675066-8
[19] Triantafillou, G.: Rationalization of HopfG-spaces. Math. Z.182, 485-500 (1983) · Zbl 0518.55008 · doi:10.1007/BF01215478
[20] Triantafillou, G.: An algebraic model forG-homotopy types. Astérisque113-114, 312-337 (1984)
[21] Wall, C.T.C.: Norms of units in group rings. Proc. London Math. Soc.29, 593-632 (1974) · Zbl 0302.16013 · doi:10.1112/plms/s3-29.4.593
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