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Mat. Sb., 2000, Volume 191, Number 9, Pages 3–22 (Mi msb504)  

This article is cited in 26 scientific papers (total in 26 papers)

On the index of $G$-spaces

A. Yu. Volovikov


Abstract: With a $G$-space, where $G$ is a compact Lie group, one can associate an ideal in the cohomology ring of the classifying space for $G$. It is called the ideal-valued index of the $G$-space. A filtration of the ideal-valued index that arises in a natural way from the Leray spectral sequence is considered. Properties of the index with filtration are studied and numerical indices are introduced. These indices are convenient for estimates of the $G$-category and the study of the set of critical points of a $G$-invariant functional defined on a manifold.
A generalization of the Bourgin–Yang theorem for the index with filtration is proved. This result is used for estimates of the index of the space of partial coincidences for a map of a space with $p$-torus action in a Euclidean space.

DOI: https://doi.org/10.4213/sm504

Full text: PDF file (336 kB)
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English version:
Sbornik: Mathematics, 2000, 191:9, 1259–1277

Bibliographic databases:

UDC: 515.142.226
MSC: Primary 57S10, 55R35, 55M30, 55N91; Secondary 55M20, 58E05
Received: 21.10.1999

Citation: A. Yu. Volovikov, “On the index of $G$-spaces”, Mat. Sb., 191:9 (2000), 3–22; Sb. Math., 191:9 (2000), 1259–1277

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. Yu. Volovikov, “Coincidence points of functions from $\mathbb Z_p^k$-spaces to $CW$-complexes”, Russian Math. Surveys, 57:1 (2002), 170–172  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. S. A. Bogatyi, “Borsuk's Conjecture, Ryshkov Obstruction, Interpolation, Chebyshev Approximation, Transversal Tverberg's Theorem, and Problems”, Proc. Steklov Inst. Math., 239 (2002), 55–73  mathnet  mathscinet  zmath
    3. A. Yu. Volovikov, “Equivariant Maps and Some Problems of the Geometry of Convex Sets”, Proc. Steklov Inst. Math., 239 (2002), 74–87  mathnet  mathscinet  zmath
    4. Gonçalves D.L., Jaworowski J., Pergher P.L.Q., Volovikov A.Yu., “Coincidences for maps of spaces with finite group actions”, Topology Appl., 145:1-3 (2004), 61–68  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    5. A. Yu. Volovikov, “Coincidence points of maps of $\mathbb Z_p^n$-spaces”, Izv. Math., 69:5 (2005), 913–962  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    6. A. Yu. Volovikov, “The genus of $G$-spaces and topological lower bounds for chromatic numbers of hypergraphs”, J. Math. Sci., 144:5 (2007), 4387–4397  mathnet  crossref  mathscinet  zmath  elib  elib
    7. A. Yu. Volovikov, “On the Cohen–Lusk theorem”, J. Math. Sci., 159:6 (2009), 790–793  mathnet  crossref  mathscinet  zmath  elib  elib
    8. R. N. Karasev, “Topological methods in combinatorial geometry”, Russian Math. Surveys, 63:6 (2008), 1031–1078  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    9. Bogatyi S.A., “Finite-to-one maps”, Topology Appl., 155:17-18 (2008), 1876–1887  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    10. Karasev R.N., “The genus and the category of configuration spaces”, Topology Appl., 156:14 (2009), 2406–2415  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    11. Karasev R.N., Volovikov A.Yu, “Knaster's problem for almost $(Z_p)^k$-orbits”, Topology Appl., 157:5 (2010), 941–945  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    12. Roman Karasev, Alexey Volovikov, “Configuration-like spaces and coincidences of maps on orbits”, Algebr. Geom. Topol, 11:2 (2011), 1033  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    13. Karasev R.N., “A topological central point theorem”, Topology Appl, 159:3 (2012), 864–868  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    14. Karasev R.N., Petrov F.V., “Partitions of Nonzero Elements of a Finite Field Into Pairs”, Isr. J. Math., 192:1 (2012), 143–156  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    15. Coelho F.R.C., de Mattos D., dos Santos E.L., “On the Existence of G-Equivariant Maps”, Bull. Braz. Math. Soc., 43:3 (2012), 407–421  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    16. P.V..M. Blagojević, R.N.. Karasev, “The Schwarz genus of the Stiefel manifold”, Topology and its Applications, 2013  crossref  mathscinet  isi  scopus  scopus  scopus
    17. Deo S., “Index of a Finitistic Space and a Generalization of the Topological Central Point Theorem”, J. Ramanujan Math. Soc., 28:2 (2013), 223–232  mathscinet  zmath  isi
    18. Oleg R. Musin, Alexey Yu. Volovikov, “Borsuk–Ulam type spaces”, Mosc. Math. J., 15:4 (2015), 749–766  mathnet  mathscinet
    19. Blagojevic P.V.M., Lueck W., Ziegler G.M., “Equivariant Topology of Configuration Spaces”, J. Topol., 8:2 (2015), 414–456  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    20. Singh M., “Equivariant Maps from Stiefel Bundles to Vector Bundles”, Proc. Edinb. Math. Soc., 60:1 (2017), 231–250  crossref  mathscinet  zmath  isi  scopus
    21. Singh S.K., Singh H.K., Singh T.B., “a Borsuk-Ulam Type Theorem For the Product of a Projective Space and 3-Sphere”, Topology Appl., 225 (2017), 112–129  crossref  mathscinet  zmath  isi  scopus
    22. Singh S.K., Singh H.K., Singh T.B., “Borsuk-Ulam Theorems and Their Parametrized Versions For F P-M X S-3”, Bull. Braz. Math. Soc., 49:1 (2018), 179–197  crossref  mathscinet  zmath  isi  scopus  scopus
    23. Singh K.S., Singh H.K., Singh T.B., “Free Action of Finite Groups on Spaces of Cohomology Type (0, B)”, Glasg. Math. J., 60:3 (2018), 673–680  crossref  mathscinet  zmath  isi  scopus
    24. Blaszczyk Z. Marzantowicz W. Singh M., “General Bourgin-Yang Theorems”, Topology Appl., 249 (2018), 112–126  crossref  mathscinet  zmath  isi  scopus
    25. Baralic D., Blagojevic P.V.M., Karasev R., Vucic A., “Index of Grassmann Manifolds and Orthogonal Shadows”, Forum Math., 30:6 (2018), 1539–1572  crossref  mathscinet  zmath  isi  scopus
    26. Singh H.K. Singh K.S., “Indices of a Finitistic Space With Mod 2 Cohomology P-N X <Mml:Msup>S2</Mml:Msup>”, Indian J. Pure Appl. Math., 50:1 (2019), 23–34  crossref  mathscinet  isi  scopus
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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