Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-23T17:19:24.388Z Has data issue: false hasContentIssue false
  • English
  • Français

Betti numbers of fixed point sets and multiplicities of indecomposable summands

Part of: Representation theory of groups Classical Topics in Algebraic Topology

Published online by Cambridge University Press:  09 April 2009

Semra Öztürk Kaptanoglu
Semra Öztürk Kaptanoglu
Affiliation:
Mathematics Department Middle East Technical UniversityAnkara 06531Turkey e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let G be a finite group of even order, k be a field of characteristic 2, and M be a finitely generated kG-module. If M is realized by a compact G-Moore space X, then the Betti numbers of the fixed point set XCn and the multiplicities of indecomposable summands of M considered as a kCn-module are related via a localization theorem in equivariant cohomology, where Cn is a cyclic subgroup of G of order n. Explicit formulas are given for n = 2 and n = 4.

Keywords

Betti numberfixed point setMoore spacerealizable modulemultiplicityindecomposable summandlocalizationequivariant cohomology.

MSC classification

Secondary: 55M35: Finite groups of transformations (including Smith theory) 20C05: Group rings of finite groups and their modules
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 74 , Issue 2 , April 2003 , pp. 165 - 172
Copyright
Copyright © Australian Mathematical Society 2003

References

[AlEv]Alperin, J. L. and Evens, L., ‘Representations, resolutions, and Quillen's dimension theorem’, J. Pure Appl. Algebra 144 (1981), 1–9.CrossRefGoogle Scholar
[Ar]Arnold, J. E., ‘On Steenrod's problem for cyclic p-groups’, Canad. J. Math. 29 (1977), 421–428.CrossRefGoogle Scholar
[As1]Assadi, A. H., ‘Varieties in finite transformation groups’, Bull. Amer. Math. Soc. 19 (1998), 459–463.CrossRefGoogle Scholar
[As2]Assadi, A. H., Homotopy actions and cohomology of finite transformation groups, Lecture Notes in Math. 1217 (Springer, Berlin, 1986), pp. 26–57.Google Scholar
[As3]Assadi, A. H., ‘Algebraic geometric invariants for homotopy actions’, in: Prospects in topology (Princeton, 1994), Ann. of Math. Stud. 138 (Princeton Univ. Press, Princeton, 1995) pp. 13–27.Google Scholar
[BeHa]Benson, D. and Habbager, N., ‘Varieties for modules and a problem of Steenrod’, J. Pure Appl. Algebra 44 (1987), 13–34.CrossRefGoogle Scholar
[Ca]Carlson, J. F., ‘The variety and the cohomology ring of a module’, J. Algebra 85 (1983), 104–143.CrossRefGoogle Scholar
[Ch]Chouinard, L., ‘Projectivity and relative projectivity for group rings’, J. Pure Appl. Algebra 7 (1976), 287–302.CrossRefGoogle Scholar
[Cn]Chen, M., ‘The Postnikov tower and the Steenrod problem’, Proc. Amer. Math. Soc. 129 (2001), 1825–1831.CrossRefGoogle Scholar
[Cs]Carlsson, G., ‘A counterexample to a conjecture of Steenrod’, Invent. Math. 64 (1981), 171–174.CrossRefGoogle Scholar
[DW]Dwyer, W. G. and Wilkerson, C. W., ‘Smith theory revisited’, Ann. of Math. (2) 127 (1988), 191–198.CrossRefGoogle Scholar
[Hi]Higman, D. G., ‘Indecomposable representations at characteristic p’, Duke Math. J. 21 (1954), 377–381.CrossRefGoogle Scholar
[Hs]Hsiang, W. Y., Cohomology theory of topological transformation groups (Springer, Berlin, 1975).CrossRefGoogle Scholar
[Ka]Kaptanoglu, S. Ö., ‘A detection theorem for kZ2 × Z 4-modules via shifted cyclic subgroups’, (preprint).Google Scholar
[Qu]Quillen, D., ‘The spectrum of an equivariant cohomology ring I, II’, Ann. of Math. (2) 94 (1971), 549–602.CrossRefGoogle Scholar
[Se]Serre, J. P., ‘Sur la dimension cohomologique des groupes profinis’, Topology 3 (1965), 413–420.CrossRefGoogle Scholar
[Vo]Vogel, P., ‘A solution to the Steenrod problem for G-Moore spaces’, K-Theory 1 (1987), 325–335.CrossRefGoogle Scholar
[W]Wheeler, W. W., ‘The generic module theory’, J. Algebra 183 (1996), 205–228.CrossRefGoogle Scholar