×

Injective models of \(G\)-disconnected simplicial sets. (English) Zbl 0886.55012

The author extends the rational homotopy theory on \(G\)-disconnected simplicial sets being not necessary of finite \(G\)-type. Instead of the orbit category \(O(G)\), in this paper the author works over the category \(O(G,X)\) with one object for each component of each fixed point simplicial subset \(X^H\) of a \(G\)-simplicial set \(X\) for all subgroups \(H\) in \(G\). First he investigates the category \(kI\)-Mod of covariant functors on a small category \(I\) to the category of \(k\)-modules over a field \(k\) and presents some considerations about injective objects in the category \(kI\)-Mod. Then he extends these on the category \(I\)-\(DGA_k\) of functors from an \(EI\)-category \(I\) (all endomorphisms are isomorphisms) to the category \(DGA_k\). He presents the existence of an injective minimal model for a complete injective \(kI\)-algebra \(A\) for an \(EI\)-category \(I\). He shows that on de Rham algebra \(A^*_X\) of rational polynomial forms on a simplicial set \(X\) there is a natural complete linear topology. Finally, for a \(G\)-simplicial set \(X\) he associates the de Rham \(\mathbb{Q} O(G,X)\)-algebra \(A^*_X\) \((\mathbb{Q}\) the field of rationals) and shows the existence of an injective minimal model for the de Rham \(\mathbb{Q} O(G,X)\)-algebra \(A^*_X\). The main result describes the rational homotopy type of a nilpotent \(G\)-simplicial set \(X\) by means of injective minimal model of the de Rham \(\mathbb{Q} O(G,X)\)-algebra \(A^*_X\).

MSC:

55P62 Rational homotopy theory
55P91 Equivariant homotopy theory in algebraic topology
16W80 Topological and ordered rings and modules
18G30 Simplicial sets; simplicial objects in a category (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] [1] and , On PL de Rham theory and rational homotopy type, Memories Amer. Math. Soc., 179 (1976). · Zbl 0338.55008
[2] [2] , Equivariant Cohomology Theories, Lecture Notes in Math., Springer-Verlag, 34 (1967). · Zbl 0162.27202
[3] [3] , System of fixed point sets, Trans. Amer. Math. Soc., 277 (1983), 275-284. · Zbl 0521.57027
[4] [24] , Good properties of algebras of invariants and defect of linear representations, Journal of Lie Theory, 5 (1995
[5] [5] and , On the equivariant formality of Kähler manifolds with finite group action, Can. J. Math., 45 (1993), 1200-1210. · Zbl 0805.55009
[6] [6] , Injectivity of the de Rham algebra on G-disconnected simplicial sets, (submitted).
[7] [7] , Equivariant Rational Homotopy Theory as a Closed Model Category, J. Pure Appl. Alg., (to appear). · Zbl 0929.55012
[8] [8] , Componentwise injective models of functors to DGAs, Colloq. Math., 73 (1997), 83-92. · Zbl 0877.55004
[9] [9] , Lectures on minimal models, Mémories S.M.F., nouvelle série, 9-10 (1983). · Zbl 0536.55003
[10] [10] , Algebraic topology, Amer. Math. Soc. Colloq. Publ., XXVII (1942). · Zbl 0061.39302
[11] [11] , Transformation groups and Algebraic K-Theory, Lect. Notes in Math., Springer-Verlag, 1408 (1989). · Zbl 0679.57022
[12] [12] , Infinitesimal Computations in Topology, Publ. Math. I.H.E.S., 47 (1977), 269-331. · Zbl 0374.57002
[13] [13] , Equivariant minimal models, Trans. Amer. Math. Soc., 274 (1982), 509-532. · Zbl 0516.55010
[14] [14] , Ratinalization of Hopf G-spaces, Math. Z., 182 (1983), 485-500. · Zbl 0518.55008
[15] [15] , An algebraic model for G-homotopy types, Astérisque, 113-114 (1984), 312-337. · Zbl 0564.55009
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.