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Russian Mathematical Surveys, 2017, Volume 72, Issue 1, Pages 1–32
DOI: https://doi.org/10.1070/RM9748
(Mi rm9748)
 

This article is cited in 10 scientific papers (total in 10 papers)

Equivariant analogues of the Euler characteristic and Macdonald type equations

S. M. Gusein-Zade

Moscow State University
References:
Abstract: One of the simplest and, at the same time, most important invariants of a topological space is the Euler characteristic. A generalization of the notion of the Euler characteristic to the equivariant setting, that is, to spaces with an action of a group (say, finite) is far from unique. An equivariant analogue of the Euler characteristic can be defined as an element of the ring of representations of the group or as an element of the Burnside ring of the group. From physics came the notion of the orbifold Euler characteristic, and this was generalized to orbifold Euler characteristics of higher orders. The main property of the Euler characteristic (defined in terms of the cohomology with compact support) is its additivity. On some classes of spaces there are additive invariants other than the Euler characteristic, and they can be regarded as generalized Euler characteristics. For example, the class of a variety in the Grothendieck ring of complex quasi-projective varieties is a universal additive invariant on the class of complex quasi-projective varieties. Generalized analogues of the Euler characteristic can also be defined in the equivariant setting. There is a simple formula — the Macdonald equation — for the generating series of the Euler characteristics of the symmetric powers of a space: it is equal to the series $(1-t)^{-1}=1+t+t^2+\cdots$ independent of the space, raised to a power equal to the Euler characteristic of the space itself. Equations of a similar kind for other invariants (‘equivariant and generalized Euler characteristics’) are called Macdonald type equations. This survey discusses different versions of the Euler characteristic in the equivariant setting and describes some of their properties and Macdonald type equations.
Bibliography: 59 titles.
Keywords: finite group actions, equivariant Euler characteristic, orbifold Euler characteristic.
Funding agency Grant number
Russian Science Foundation 16-11-10018
This work was supported by the Russian Science Foundation under grant 16-11-10018.
Received: 16.10.2016
Revised: 13.12.2016
Bibliographic databases:
Document Type: Article
UDC: 515.171.5+515.165
MSC: Primary 57S17, 57R20; Secondary 32M99, 32Q55
Language: English
Original paper language: Russian
Citation: S. M. Gusein-Zade, “Equivariant analogues of the Euler characteristic and Macdonald type equations”, Russian Math. Surveys, 72:1 (2017), 1–32
Citation in format AMSBIB
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\by S.~M.~Gusein-Zade
\paper Equivariant analogues of the Euler characteristic and Macdonald type equations
\jour Russian Math. Surveys
\yr 2017
\vol 72
\issue 1
\pages 1--32
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  • This publication is cited in the following 10 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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