×

Equivariant cohomology algebras of \(\mathfrak{F}\)-classifying \(T^k\)-spaces. (Algebras of the equivariant cohomologies of an \(\mathfrak{F}\)-classifying \(T^k\)-spaces.) (English. Russian original) Zbl 1365.55004

Russ. Math. 59, No. 1, 51-59 (2015); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2015, No. 1, 60-70 (2015).
The author considers the toral group \(G= T^k\). For any \(G\) space \(X,\) let \(\mathcal{F}\) denote the family of stabilizer subgroups \(G_x,~x\in X\). The author assumes that \(\mathcal{F}\) is a finite collection of finite groups.
The construction of universal \(G\)-bundles was generalised by Palais in 1960 to Universal \(G\)-spaces. In this paper, \(\mathbb{W}\) denotes the Universal Palais \(G\)-space or its variant \(\mathbb{W}_{\mathcal{F}}\) for the family \(\mathcal{F}\). Let \(W_{\mathcal{F}}= \mathbb{W}_{\mathcal{F}}/G\) and denote \[ E_{\mathcal{F}} X= \{(w,x)\in \mathbb{W}\times X: ~ G_w\subset G_x, ~(G_w)\in \mathcal{F} \}. \] Then define \(H_{\mathcal{F}}^*(X)= H^*(E_{\mathcal{F}} X; \mathbb{Q})\). A family \(\mathcal{F}\) is said to be typical if \(H^*(W_{\mathcal{F}}; \mathbb{Q})= \mathbb{Q}[x_1,x_2,\dots, x_k]\), where each \(x_i\) is of degree 2. The first result says that if \({\mathcal{F}}_1, {\mathcal{F}}_2, ~{\mathcal{F}}_0= {\mathcal{F}}_1\cap {\mathcal{F}}_2\) are typical then \({\mathcal{F}}={\mathcal{F}}_1\cup {\mathcal{F}}_2\) is typical. Two other theorems describe how the equivariant cohomology \(H^*_{\mathcal{F}}(\mathbb{W}_{\mathcal{F}})\) depends on the equivariant cohomologies of \(\mathcal{F}_0, \mathcal{F}_1,\mathcal{F}_2\).

MSC:

55N91 Equivariant homology and cohomology in algebraic topology
57S10 Compact groups of homeomorphisms
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ageev, S.M., Repoš, D. “The Covering Homotopy Extension Problem for Compact Transformation Groups,” Math. Notes 92, No. 5-6, 737-750 (2012). · Zbl 1271.54057 · doi:10.1134/S0001434612110181
[2] Palais, R. “The Classification of <Emphasis Type=”Italic“>G-spaces,” Mem. Amer. Math. Soc. 36 (1960). · Zbl 0119.38403
[3] Ageev, S.M. “On Palais Universal <Emphasis Type=”Italic“>G-Spaces and Isovariant Absolute Extensors,” Sb. Math. 203, No. 5-6, 769-797 (2012). · Zbl 1257.54038 · doi:10.1070/SM2012v203n06ABEH004242
[4] Ageev, S. M. “Isovariant Extensors and the Characterization of Equivariant Homotopy Equivalences,” Izv. Math. 76, No. 5, 857-880 (2012). · Zbl 1266.54081 · doi:10.1070/IM2012v076n05ABEH002607
[5] Hsiang, Wu Yi. Cohomology Theory of Topological Transformation Goups (Springer-Verlag, 1975; Mir, Moscow, 1979). · Zbl 0429.57011 · doi:10.1007/978-3-642-66052-8
[6] Ageev, S. M., Usimov, I. V. “The Cohomology Ring of Subspace of Universal <Emphasis Type=”Italic“>S1-Space with Finite Orbit Types,” Topology Appl. 160, No. 11, 1255-1260 (2013). · Zbl 1296.55010 · doi:10.1016/j.topol.2013.04.018
[7] Bredon, G. E. Equivariant Cohomology Theories, Lecture Notes in Math., No. 34 (Springer-Verlag, Berlin-New York, 1967). · Zbl 0162.27202
[8] Fomenko, A. T, Fuks, D. B. Homotopic Topology (Nauka, Moscow, 1989) [in Russian]. · Zbl 0675.55001
[9] Morris, S. A. Pontryagin Duality and the Structure of Locally Compact Abelian Groups (Cambridge University Press, 1977; Mir, Moscow, 1980). · Zbl 0446.22006 · doi:10.1017/CBO9780511600722
[10] Bott, R., Tu, L. W. Differential Forms in Algebraic Topology (Springer-Verlag, 1982; Nauka, Moscow, 1989). · Zbl 0496.55001 · doi:10.1007/978-1-4757-3951-0
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.