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Marek Golasiński
On $G$-disconnected injective models
(Sur les $G$-modèles injectifs non connexess)
Annales de l'institut Fourier, 53 no. 2 (2003), p. 625-664, doi: 10.5802/aif.1954
Article PDF | Reviews MR 1990008 | Zbl 01940706
Class. Math.: 55P62, 55P91, 16W80, 18G30
Keywords: differential graded algebra, de Rham algebra, $EI$-category, $i$-elementary extension, $i$-minimal model, linearly compact (complete) $k$-module, Postnikov tower, quasi-isomorphism, rationalization, $G$-simplicial set
Let $G$ be a finite group. It was observed by L.S. Scull that the original definition of
the equivariant minimality in the $G$-connected case is incorrect because of an error
concerning algebraic properties. In the $G$-disconnected case the orbit category ${\cal
O}(G)$ was originally replaced by the category ${\cal O}(G,X)$ with one object for each
component of each fixed point simplicial subsets $X^H$ of a $G$-simplicial set $X$, for
all subgroups $H\subseteq G$. We redefine the equivariant minimality and redevelop some
results on the rational homotopy theory of disconnected $G$-simplicial sets. To show an
existence of the injective minimal model ${\cal M}_X$ for a disconnected $G$-simplicial
set $X$ we replace ${\cal O}(G,X)$ by the more subtle category $\tilde{\cal O}(G,X)$ with
one object for each 0-simplex of fixed point simplicial subsets $X^H$, for all subgroups
$H\subseteq G$.
[1] A.K. Bousfield & V.K.A.M. Gugenheim, “On PL de Rham theory and rational homotopy type”, Memories of the Amer. Math. Soc. 179 (1976) MR 425956 | Zbl 0338.55008
[2] G. Bredon, “Equivariant Cohomology Theories”, Lecture Notes in Math vol. 34 (1967) MR 206946 | Zbl 0162.27202
[3] J.-M. Cordier & T. Porter, “Homotopy coherent category theory”, Trans. Amer. Math. Soc. 349 (1997) no. 1, p. 1-54 Article | MR 1376543 | Zbl 0865.18006
[4] P. Deligne, P. Griffiths, J. Morgan & D. Sullivan, “Real homotopy theory of Kähler manifolds”, Invent. Math. vol. 29 (1975), p. 245-274 Article | MR 382702 | Zbl 0312.55011
[5] B.L. Fine, “Disconnected equivariant rational homotopy theory and formality of compact G-Kähler manifolds”, Ph.D. thesis, Chicago, 1992
[6] B.L. Fine & G.V. Triantafillou, “On the equivariant formality of Kähler manifolds with finite group action”, Can. J. Math. 45 (1993), p. 1200-1210 Article | MR 1247542 | Zbl 0805.55009
[7] M. Golasiński, “Injective models of G-disconnected simplicial sets”, Ann. Inst. Fourier, Grenoble 47 (1997) no. 5, p. 1491-1522 Cedram | MR 1600367 | Zbl 0886.55012
[8] M. Golasiński, “Injectivity of functors to modules and $DGA's$”, Comm. Algebra 27 (1999), p. 4027-4038 Article | MR 1700197 | Zbl 0942.18005
[9] M. Golasiński, “On the object-wise tensor product of functors to modules”, Theory Appl. Categ. 7 (2000) no. 1, p. 227-235 MR 1779432 | Zbl 0965.18008
[10] M. Golasiński, “Component-wise injective models of functors to $DGAs$”, Colloq. Math. 73 (1997), p. 83-92 MR 1436952 | Zbl 0877.55004
[11] S. Halperin, “Lectures on minimal models”, Mémories S.M.F., Nouvelle série (1983), p. 9-10 Numdam | Zbl 0536.55003
[12] P. Hilton, G. Mislin & J. Roitberg, Localization of Nilpotent Groups and Spaces, Noth-Holland Mathematics Studies 15, Amsterdam, 1975 MR 478146 | Zbl 0323.55016
[13] S. Lefschetz, “Algebraic Topology”, Amer. Math. Soc. Colloq. Publ. XXVII (1942) Zbl 0061.39302
[14] D. Lehmann, Théorie homotopique des formes différentielles, Astérique 45, S.M.F., 1977 Zbl 0367.55008
[15] W. Lück, Transformation groups and Algebraic K-Theory, Lect. Notes in Math. 1408, Springer-Verlag, 1989 MR 1027600 | Zbl 0679.57022
[16] J.P. May, Simplicial Objects in Algebraic Topology, Van Nostrand Mathematical Studies 11, Princeton-Toronto-London-Melbourne, 1967 MR 222892 | Zbl 0165.26004
[17] L.S. Scull, “Rational $\Bbb S^\eightpoint 1$-equivariant homotopy theory”, Trans. Amer. Math. Soc. 354 (2002), p. 1-45 Article | MR 1859023 | Zbl 0989.55009
[18] D. Sullivan, “Infinitesimal Computations in Topology”, Publ. Math. I.H.E.S. 47 (1977), p. 269-331 Numdam | MR 646078 | Zbl 0374.57002
[19] G.V. Triantafillou, “Equivariant minimal models”, Trans. Amer. Math. Soc. 274 (1982), p. 509-532 Article | MR 675066 | Zbl 0516.55010
[20] G.V. Triantafillou, “An algebraic model for G-homotopy types”, Astérisque 113-114 (1984), p. 312-337 MR 749073 | Zbl 0564.55009
Marek Golasiński
On $G$-disconnected injective models
(Sur les $G$-modèles injectifs non connexess)
Annales de l'institut Fourier, 53 no. 2 (2003), p. 625-664, doi: 10.5802/aif.1954
Article PDF | Reviews MR 1990008 | Zbl 01940706
Class. Math.: 55P62, 55P91, 16W80, 18G30
Keywords: differential graded algebra, de Rham algebra, $EI$-category, $i$-elementary extension, $i$-minimal model, linearly compact (complete) $k$-module, Postnikov tower, quasi-isomorphism, rationalization, $G$-simplicial set
Résumé - Abstract
Bibliography
[2] G. Bredon, “Equivariant Cohomology Theories”, Lecture Notes in Math vol. 34 (1967) MR 206946 | Zbl 0162.27202
[3] J.-M. Cordier & T. Porter, “Homotopy coherent category theory”, Trans. Amer. Math. Soc. 349 (1997) no. 1, p. 1-54 Article | MR 1376543 | Zbl 0865.18006
[4] P. Deligne, P. Griffiths, J. Morgan & D. Sullivan, “Real homotopy theory of Kähler manifolds”, Invent. Math. vol. 29 (1975), p. 245-274 Article | MR 382702 | Zbl 0312.55011
[5] B.L. Fine, “Disconnected equivariant rational homotopy theory and formality of compact G-Kähler manifolds”, Ph.D. thesis, Chicago, 1992
[6] B.L. Fine & G.V. Triantafillou, “On the equivariant formality of Kähler manifolds with finite group action”, Can. J. Math. 45 (1993), p. 1200-1210 Article | MR 1247542 | Zbl 0805.55009
[7] M. Golasiński, “Injective models of G-disconnected simplicial sets”, Ann. Inst. Fourier, Grenoble 47 (1997) no. 5, p. 1491-1522 Cedram | MR 1600367 | Zbl 0886.55012
[8] M. Golasiński, “Injectivity of functors to modules and $DGA's$”, Comm. Algebra 27 (1999), p. 4027-4038 Article | MR 1700197 | Zbl 0942.18005
[9] M. Golasiński, “On the object-wise tensor product of functors to modules”, Theory Appl. Categ. 7 (2000) no. 1, p. 227-235 MR 1779432 | Zbl 0965.18008
[10] M. Golasiński, “Component-wise injective models of functors to $DGAs$”, Colloq. Math. 73 (1997), p. 83-92 MR 1436952 | Zbl 0877.55004
[11] S. Halperin, “Lectures on minimal models”, Mémories S.M.F., Nouvelle série (1983), p. 9-10 Numdam | Zbl 0536.55003
[12] P. Hilton, G. Mislin & J. Roitberg, Localization of Nilpotent Groups and Spaces, Noth-Holland Mathematics Studies 15, Amsterdam, 1975 MR 478146 | Zbl 0323.55016
[13] S. Lefschetz, “Algebraic Topology”, Amer. Math. Soc. Colloq. Publ. XXVII (1942) Zbl 0061.39302
[14] D. Lehmann, Théorie homotopique des formes différentielles, Astérique 45, S.M.F., 1977 Zbl 0367.55008
[15] W. Lück, Transformation groups and Algebraic K-Theory, Lect. Notes in Math. 1408, Springer-Verlag, 1989 MR 1027600 | Zbl 0679.57022
[16] J.P. May, Simplicial Objects in Algebraic Topology, Van Nostrand Mathematical Studies 11, Princeton-Toronto-London-Melbourne, 1967 MR 222892 | Zbl 0165.26004
[17] L.S. Scull, “Rational $\Bbb S^\eightpoint 1$-equivariant homotopy theory”, Trans. Amer. Math. Soc. 354 (2002), p. 1-45 Article | MR 1859023 | Zbl 0989.55009
[18] D. Sullivan, “Infinitesimal Computations in Topology”, Publ. Math. I.H.E.S. 47 (1977), p. 269-331 Numdam | MR 646078 | Zbl 0374.57002
[19] G.V. Triantafillou, “Equivariant minimal models”, Trans. Amer. Math. Soc. 274 (1982), p. 509-532 Article | MR 675066 | Zbl 0516.55010
[20] G.V. Triantafillou, “An algebraic model for G-homotopy types”, Astérisque 113-114 (1984), p. 312-337 MR 749073 | Zbl 0564.55009
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