Generalized Artin and Brauer induction for compact Lie groups
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Abstract:
Let $G$ be a compact Lie group. We present two induction theorems for certain generalized $G$-equivariant cohomology theories. The theory applies to $G$-equivariant $K$-theory $K_G$, and to the Borel cohomology associated with any complex oriented cohomology theory. The coefficient ring of $K_G$ is the representation ring $R(G)$ of $G$. When $G$ is a finite group the induction theorems for $K_G$ coincide with the classical Artin and Brauer induction theorems for $R(G)$.References
- M. F. Atiyah and G. B. Segal, Equivariant $K$-theory and completion, J. Differential Geometry 3 (1969), 1–18. MR 259946
- D. J. Benson, Representations and cohomology. I, Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1991. Basic representation theory of finite groups and associative algebras. MR 1110581
- Agnieszka Bojanowska, The spectrum of equivariant $K$-theory, Math. Z. 183 (1983), no. 1, 1–19. MR 701356, DOI 10.1007/BF01187212
- Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
- M. Brun, Witt vectors and equivariant ring spectra applied to cobordism, Proc. Lond. Math. Soc. (3) 94 (2007), no. 2, 351–385. MR 2308231, DOI 10.1112/plms/pdl010
- S. B. Conlon, Decompositions induced from the Burnside algebra, J. Algebra 10 (1968), 102–122. MR 237664, DOI 10.1016/0021-8693(68)90107-5
- Tammo tom Dieck, Bordism of $G$-manifolds and integrality theorems, Topology 9 (1970), 345–358. MR 266241, DOI 10.1016/0040-9383(70)90058-3
- Tammo tom Dieck, The Burnside ring of a compact Lie group. I, Math. Ann. 215 (1975), 235–250. MR 394711, DOI 10.1007/BF01343892
- Tammo tom Dieck, A finiteness theorem for the Burnside ring of a compact Lie group, Compositio Math. 35 (1977), no. 1, 91–97. MR 474344
- Tammo tom Dieck, Transformation groups and representation theory, Lecture Notes in Mathematics, vol. 766, Springer, Berlin, 1979. MR 551743
- Halvard Fausk and Bob Oliver, Continuity of $\pi$-perfection for compact Lie groups, Bull. London Math. Soc. 37 (2005), no. 1, 135–140. MR 2106728, DOI 10.1112/S0024609304003601
- Mark Feshbach, The transfer and compact Lie groups, Trans. Amer. Math. Soc. 251 (1979), 139–169. MR 531973, DOI 10.1090/S0002-9947-1979-0531973-8
- Mark Feshbach, Some general theorems on the cohomology of classifying spaces of compact Lie groups, Trans. Amer. Math. Soc. 264 (1981), no. 1, 49–58. MR 597866, DOI 10.1090/S0002-9947-1981-0597866-4
- J. P. C. Greenlees, Equivariant connective $K$-theory for compact Lie groups, J. Pure Appl. Algebra 187 (2004), no. 1-3, 129–152. MR 2027899, DOI 10.1016/j.jpaa.2003.07.008
- Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
- Michael J. Hopkins, Nicholas J. Kuhn, and Douglas C. Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (2000), no. 3, 553–594. MR 1758754, DOI 10.1090/S0894-0347-00-00332-5
- J. P. C. Greenlees and J. P. May, Generalized Tate cohomology, Mem. Amer. Math. Soc. 113 (1995), no. 543, viii+178. MR 1230773, DOI 10.1090/memo/0543
- Stefan Jackowski, Equivariant $K$-theory and cyclic subgroups, Transformation groups (Proc. Conf., Univ. Newcastle upon Tyne, Newcastle upon Tyne, 1976) London Math. Soc. Lecture Note Series, No. 26, Cambridge Univ. Press, Cambridge, 1977, pp. 76–91. MR 0448377
- L. Gaunce Lewis Jr., The category of Mackey functors for a compact Lie group, Group representations: cohomology, group actions and topology (Seattle, WA, 1996) Proc. Sympos. Pure Math., vol. 63, Amer. Math. Soc., Providence, RI, 1998, pp. 301–354. MR 1603183, DOI 10.1090/pspum/063/1603183
- L. G. Lewis Jr., J. P. May, M. Steinberger, and J. E. McClure, Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, Berlin, 1986. With contributions by J. E. McClure. MR 866482, DOI 10.1007/BFb0075778
- J. P. May, Equivariant homotopy and cohomology theory, CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996. With contributions by M. Cole, G. Comezaña, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. MR 1413302, DOI 10.1090/cbms/091
- J. P. May, Picard groups, Grothendieck rings, and Burnside rings of categories, Adv. Math. 163 (2001), no. 1, 1–16. MR 1867201, DOI 10.1006/aima.2001.1996
- James E. McClure, Restriction maps in equivariant $K$-theory, Topology 25 (1986), no. 4, 399–409. MR 862427, DOI 10.1016/0040-9383(86)90019-4
- Deane Montgomery and Leo Zippin, A theorem on Lie groups, Bull. Amer. Math. Soc. 48 (1942), 448–452. MR 6545, DOI 10.1090/S0002-9904-1942-07699-3
- Goro Nishida, The transfer homomorphism in equivariant generalized cohomology theories, J. Math. Kyoto Univ. 18 (1978), no. 3, 435–451. MR 509493, DOI 10.1215/kjm/1250522505
- Bob Oliver, The representation ring of a compact Lie group revisited, Comment. Math. Helv. 73 (1998), no. 3, 353–378. MR 1633351, DOI 10.1007/s000140050059
- Graeme Segal, The representation ring of a compact Lie group, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 113–128. MR 248277
- Graeme Segal, Equivariant $K$-theory, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 129–151. MR 234452
- Louis Solomon, The Burnside algebra of a finite group, J. Combinatorial Theory 2 (1967), 603–615. MR 214679
Additional Information
- Halvard Fausk
- Affiliation: Department of Mathematics, University of Oslo, 1053 Blindern, 0316 Oslo, Norway
- Email: fausk@math.uio.no
- Received by editor(s): December 18, 2006
- Published electronically: April 14, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 360 (2008), 5043-5066
- MSC (2000): Primary 55P91, 19A22; Secondary 55P42
- DOI: https://doi.org/10.1090/S0002-9947-08-04528-5
- MathSciNet review: 2403712