Equivariant Euler-Poincaré characteristic in sheaf cohomology. (English) Zbl 1419.55008
Summary: Let \(X\) be a Hausdorff space equipped with a continuous action of a finite group \(G\) and a \(G\)-stable family of supports \(\Phi\). Fix a number field \(F\) with ring of integers \(R\). We study the class \(\chi=\sum_j(-1)^j[H^j_\Phi(X,\mathcal E)\otimes_RF]\) in the character group of \(G\) over \(F\) for any flat \(G\)-sheaf \(\mathcal E\) of \(R\)-modules over \(X\). Under natural cohomological finiteness conditions we give a formula for \(\chi\) with respect to the basis given by the irreducible characters of \(G\). We discuss applications of our result concerning the cohomology of arithmetic groups.
MSC:
| 55N30 | Sheaf cohomology in algebraic topology |
| 57R20 | Characteristic classes and numbers in differential topology |
| 55M20 | Fixed points and coincidences in algebraic topology |
| 20H05 | Unimodular groups, congruence subgroups (group-theoretic aspects) |
| 55M35 | Finite groups of transformations in algebraic topology (including Smith theory) |
References:
| [1] | Borel A., Serre J.-P.: Corners and arithmetic groups. Comment. Math. Helv. 48, 436-491 (1973) · Zbl 0274.22011 · doi:10.1007/BF02566134 |
| [2] | Bredon, G.E.: Sheaf Theory. 2nd Ed., Grad. Texts Math. (170). Springer, New York (1997) · Zbl 0874.55001 |
| [3] | Brown, K.S.: Cohomology of Groups, Grad. Texts Math. (87). Springer, New York (1982) · Zbl 0584.20036 |
| [4] | Brown K.S.: Complete Euler characteristics and fixed-point theory. J. Pure Appl. Algebra 24, 103-121 (1982) · Zbl 0493.20033 · doi:10.1016/0022-4049(82)90008-1 |
| [5] | Curtis C.W., Reiner I.: Representation Theory of Finite Groups and Associative Algebras. Interscience Publishers, New York (1962) · Zbl 0131.25601 |
| [6] | Calegari F., Emerton M.: Bounds for multiplicities of unitary representations of cohomological type in spaces of cusp forms. Ann. Math. 170(2), 1437-1446 (2009) · Zbl 1195.22015 · doi:10.4007/annals.2009.170.1437 |
| [7] | Demazure M., Gabriel P.: Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs. North-Holland Publishing Co., Amsterdam (1970) · Zbl 0203.23401 |
| [8] | Floyd E.E.: On periodic maps and the Euler characteristics of associated spaces. Trans. Am. Math. Soc. 72, 138-147 (1952) · Zbl 0046.16603 · doi:10.1090/S0002-9947-1952-0046039-4 |
| [9] | Godement, R.: Topologie Algébrique et Theorie des Faisceaux, Act. sci. ind. (1252). Hermann, Paris (1958) · Zbl 0080.16201 |
| [10] | Grothendieck A.: Sur quelques points d’algèbre homologique. Tohoku Math. J. 9(2), 119-221 (1957) · Zbl 0118.26104 · doi:10.2748/tmj/1178244839 |
| [11] | Harder G.: A Gauss-Bonnet formula for discrete arithmetically defined groups. Ann. Sci. École Norm. Sup. 4, 409-455 (1971) · Zbl 0232.20088 |
| [12] | Harder, G.: Eisensteinkohomologie und die Konstruktion gemischter Motive, Lect. Notes Math. (1562). Springer, Berlin (1993) · Zbl 0795.11024 |
| [13] | Illusie, L., Zheng, W.: Odds and ends on finite group actions and traces. Int. Math. Res. Not. 2013(1), 1-62 (2013) · Zbl 1360.14120 |
| [14] | Kionke, S.: Lefschetz Numbers of Involutions on Arithmetic Subgroups of Inner Forms of the Special Linear Group, Dissertation, Universität Wien (2012) · Zbl 0118.26104 |
| [15] | Kionke S.: On lower bounds for cohomology growth in p-adic analytic towers. Math. Z. 277, 709-723 (2014) · Zbl 1354.20032 · doi:10.1007/s00209-013-1273-3 |
| [16] | Rohlfs J.: Arithmetisch definierte Gruppen mit Galoisoperation. Invent. math. 48, 185-205 (1978) · Zbl 0391.14007 · doi:10.1007/BF01390250 |
| [17] | Rohlfs, J.: Lefschetz numbers for arithmetic groups, Cohomology of arithmetic groups and automorphic forms. In: Proc., Lect. Notes Math. (1447), pp. 303-313. Springer (1990) · Zbl 0762.11023 |
| [18] | Rohlfs J., Speh B.: On limit multiplicities of representations with cohomology in the cuspidal spectrum. Duke Math. J. 55, 199-211 (1987) · Zbl 0626.22008 · doi:10.1215/S0012-7094-87-05511-6 |
| [19] | Savin G.: Limit multiplicities of cusp forms. Invent. Math. 95, 149-159 (1989) · Zbl 0673.22003 · doi:10.1007/BF01394147 |
| [20] | Spanier E.H.: Algebraic Topology. Springer, New York (1966) · Zbl 0145.43303 |
| [21] | Verdier J.-L.: Caractéristique d’Euler-Poincaré. Bull. Soc. Math. France 101, 441-445 (1973) · Zbl 0302.57007 |
| [22] | Zarelua, A.: On finite groups of transformations. In: Proceedings of the International Symposium on Topology and Its Applications, Yugoslavia (1969) · Zbl 0211.55702 |
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.
