×

zbMATH — the first resource for mathematics

A Shapiro lemma for diagrams of spaces with applications to equivariant topology. (English) Zbl 0853.55005
The authors study equivariant Bredon (co-)homology with twisted coefficients. Their formulation of the equivariant Whitehead theorem (Theorem 1.1) allows the fixed point sets as non(simply) connected. (For a more restricted version see [S. Illman, Equivariant singular homology and cohomology I, Mem. Am. Math. Soc. 156 (1975; Zbl 0297.55003)].)
The Shapiro lemma of the title addresses an isomorphism \[ H^*(\text{induced } (X), m)\cong H^* (X, \text{restricted } (M)) \] as well as a spectral sequence \[ H^p (X, R^q (\text{induced }(M)) \Rightarrow H^{p+q} (\text{restricted }(X), (M)). \] The setting is for general diagrams of spaces. Two useful appendices recollect categorical constructions used in the paper.

MSC:
55N25 Homology with local coefficients, equivariant cohomology
55N91 Equivariant homology and cohomology in algebraic topology
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] G.E. Bredon , Equivariant Cohomology Theories , SLN 34 (1967). · Zbl 0162.27202
[2] Th. Bröcker , Singuläre Definition der äquivarianten Bredon homologie , Manus. Math. 5 (1971), 91-102. · Zbl 0213.49902 · doi:10.1007/BF01397610 · eudml:154060
[3] K.S. Brown , Cohomology of Groups , Springer-Verlag, 1982. · Zbl 0584.20036
[4] T. Tom Dieck , Transformation Groups and Representation Theory , SLN 766 (1979). · Zbl 0445.57023
[5] T. Tom Dieck , Transformation Groups , de Gruyter, 1987. · Zbl 0611.57002
[6] A. Dold , D. Puppe , Homologie nicht-additiver Funktoren. Anwendungen , Ann. de l’Inst. Fourier 11 (1961), 201-312. · Zbl 0098.36005 · doi:10.5802/aif.114 · numdam:AIF_1961__11__201_0 · eudml:73776
[7] E. Dror Farjoun , Homotopy and homology of diagrams of spaces , in SLN 1286 (1987), 93-134. · Zbl 0659.55011
[8] E. Dror Farjoun, A. Zabrodsky, Homotopy equivalence between diagrams of spaces , J. Pure Appl. Alg. 41 (1986), 169-182. · Zbl 0612.55017 · doi:10.1016/0022-4049(86)90108-8
[9] W. Dwyer , D.M. Kan , Function complexes for diagrams of simplicial sets , Proc. Kon. Ned. Acad. Wet. 86 (= Indag. Math. 45) (1983), 139-146. · Zbl 0555.55018
[10] W. Dwyer. D.M. Kan , An obstruction theory for diagrams of simplicial sets , Proc. Kon. Ned. Acad. Wet. 87 (=Indag. Math. 46) (1984), 139-149. · Zbl 0555.55018
[11] B. Eckmann , Cohomology of groups and transfers , Ann. Math. 58 (1983), 481-493. · Zbl 0052.02002 · doi:10.2307/1969749
[12] A.D. Elmendorf , Systems of fixed point sets , Trans. A.M.S. 277 (1983), 275-284. · Zbl 0521.57027 · doi:10.2307/1999356
[13] A. Grothendieck , Sur quelques points de l’algèbre homologique , Tôhoku Math J. 9 (1957), 119-221. · Zbl 0118.26104
[14] P. Gabriel , M. Zisman , Calculus of Fractions and Homotopy Theory , Springer-Verlag, 1967. · Zbl 0186.56802
[15] A. Heller , Homotopy theories , Memoirs A.M.S. 383 (1988). · Zbl 0643.55015
[16] P.J. Hilton, U. Stammbach , A Course in Homological Algebra , Springer-Verlag, 1971. · Zbl 0238.18006
[17] L. Illusie , Complexe cotangent et déformations II , SLN 283 (1972). · Zbl 0224.13014 · doi:10.1007/BFb0059052
[18] K. Lamotke , Semi-simpliziale algebraische Topologie , Springer-Verlag, 1968. · Zbl 0188.28301
[19] L.G. Lewis , J.P. May , M. Steinberger , Equivariant Stable Homotopy Theory , SLN 1213 (1986). · Zbl 0611.55001 · doi:10.1007/BFb0075778
[20] J.P. May , Simplicial Objects in Algebraic Topology , Van Nostrand, 1967 (reprinted by Univ. of Chicago Press, 1982). · Zbl 0565.55001
[21] I. Moerdijk , J.A. Svensson , The equivariant Serre spectral sequence , Proc. A.M.S. 118 (1993), 263-278. · Zbl 0791.55004 · doi:10.2307/2160037
[22] J.M. Møller , On equivariant function spaces , Pacific J. Math. 142 (1990), 103-119. · Zbl 0673.55012 · doi:10.2140/pjm.1990.142.103
[23] M.Y. Oruç , The equivariant Steenrod algebra , Top. Appl. 32 (1989), 77-108. · Zbl 0675.55010 · doi:10.1016/0166-8641(89)90008-4
[24] D. Quillen , Higher algebraic K-theory I , in SLN 341 (1973), 85-147. · Zbl 0292.18004 · doi:10.1007/BFb0067053
[25] R.M. Seymour , Some functorial constructions on G-spaces , Bull. London Math. Soc. 15 (1983), 353-359. · Zbl 0519.57035 · doi:10.1112/blms/15.4.353
[26] J S\?Omińska , On the equivariant Chern homomorphism , Bull. de l’Acad. Pol. Sci. 24 (1976), 909-913. · Zbl 0343.55003
[27] R.W. Thomason , Homotopy colimits in the category of small categories , Math. Proc. Camb. Phil. Soc. 85 (1979), 91-109. · Zbl 0392.18001 · doi:10.1017/S0305004100055535
[28] F. Waldhausen , Algebraic K-theory of spaces , in SLN 1126 (1985), 318-419. · Zbl 0579.18006
[29] A. Weil , Sur la théorie du corps de classes , J. Math. Soc. Japan 3 (1951), 1-35 (see footnote p 12). · Zbl 0044.02901 · doi:10.2969/jmsj/00310001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.