Available in electronic format
Available in print format		 Transacrions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article
     

Generalized Artin and Brauer induction for compact Lie groups

Author(s): Halvard Fausk
Journal: Trans. Amer. Math. Soc. 360 (2008), 5043-5066.
MSC (2000): Primary 55P91, 19A22; Secondary 55P42
Posted: April 14, 2008
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

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:

[AS69]
M.F. Atiyah and G.B. Segal, Equivariant $ K$-theory and completion, J. Differential Geometry 3, 1-18, 1969. MR 0259946 (41:4575)

[Ben95]
D.J. Benson, Representations and cohomology I, Cambridge Studies in Advanced Mathematics no. 30, 1991. MR 1110581 (92m:20005)

[Boj83]
A. Bojanowska, The spectrum of equivariant $ K$-theory, Math. Z. 183, no. 1, 1-19, 1983. MR 701356 (85a:55003)

[Bre72]
G.E. Bredon, Introduction to compact transformation groups, Academic Press, 1972. MR 0413144 (54:1265)

[Bru04]
M. Brun, Witt Vectors and Equivariant Ring Spectra applied to cobordism, Proc. London Math. Soc. 94, no. 2, 351-385, 2007. MR 2308231 (2008a:55008)

[Con68]
S.B. Conlon, Decompositions induced from the Burnside algebra, J. Algebra 10, 102-122, 1968. MR 0237664 (38:5945)

[tD70]
T. tom Dieck, Bordism of $ G$-manifolds and integrality theorems, Topology 9, 345-358, 1970. MR 0266241 (42:1148)

[tD75]
T. tom Dieck, The Burnside ring of a compact Lie group I, Math. Ann. 215, 235-250, 1975. MR 0394711 (52:15510)

[tD77]
T. tom Dieck, A finiteness theorem for the Burnside ring of a compact Lie group, Compositio Math. 35, 91-97, 1977. MR 0474344 (57:13990)

[tD79]
T. tom Dieck, Transformation groups and representation theory, Lecture Notes in Math., Vol. 766. Springer-Verlag. 1979. MR 551743 (82c:57025)

[FO05]
H. Fausk and B. Oliver, Continuity of $ \pi$-perfection for compact Lie groups, Bull. London Math. Soc. 37, 135-140, 2005. MR 2106728 (2005g:55018)

[Fes79]
M. Feshbach, The transfer and compact Lie groups, Trans. Amer. Math. Soc. 251, 139-169, 1979. MR 531973 (80k:55049)

[Fes81]
M. Feshbach, Some general theorems on the cohomology of classifying spaces of compact groups, Trans. Amer. Math. Soc. 264, 49-58, 1981. MR 597866 (82e:55005)

[Gre04]
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 (2004j:19007)

[Hat02]
A. Hatcher, Algebraic topology, Cambridge University Press, 2002. MR 1867354 (2002k:55001)

[HKR00]
M.J. Hopkins, N.J. Kuhn and D.C. Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13, 553-594, 2000. MR 1758754 (2001k:55015)

[GM95]
J.P.C. Greenlees and J.P. May, Generalized Tate cohomology, Memoirs of the Amer. Math. Soc. Number 543, 1995. MR 1230773 (96e:55006)

[Jac77]
S. Jackowski, Equivariant $ K$-theory and cyclic subgroups, in C. Kosniowski, Transformation groups, London Math. Soc. no. 26, 76-91, 1977. MR 0448377 (56:6684)

[Lew96]
L.G. Lewis, The category of Mackey functors for a compact Lie group, Group representations: Cohomology, group actions and topology (Seattle, WA, 1996), 301-354, Proc. Sympos. Pure Math., 63, Amer. Math. Soc., Providence, RI, 1998. MR 1603183 (99b:19001)

[LMS86]
L.G. Lewis, Jr., J.P. May, and M. Steinberger (with contributions by J.E. McClure), Equivariant stable homotopy theory, Lecture Notes in Math., 1213, Springer-Verlag, Berlin, 1986. MR 0866482 (88e:55002)

[May96]
J.P. May, Equivariant homotopy and cohomology theory, CBMS, AMS, no. 91, 1996. MR 1413302 (97k:55016)

[May01]
J.P. May. Picard groups, Grothendick rings, and Burnside rings of categories, Advances in Mathematics 163, 1-16, 2001. MR 1867201 (2002k:18011)

[McC86]
J.E. McClure, Restriction maps in equivariant $ K$-theory, Topology Vol. 25, No. 4, 399-409, 1986. MR 862427 (88f:55022)

[MZ42]
D. Montgomery and L. Zippin, Theorem on Lie groups, Bull. Amer. Math. Soc. 48, 448-552, 1942. MR 0006545 (4:3a)

[Nis78]
G. Nishida, The transfer homomorphism in equivariant generalized cohomology theories, J. Math. Kyoto Univ. 18 , no. 3, 435-451, 1978. MR 509493 (80a:55006)

[Oli98]
B. Oliver, The representation ring of a compact Lie group revisited, Comment. Mat. Helv. 73, 1998. MR 1633351 (2000e:22003)

[Seg68a]
G.B. Segal, The representation ring of a compact Lie group, Publ. Math. IHES 34, 113-128, 1968. MR 0248277 (40:1529)

[Seg68b]
G.B. Segal, Equivariant $ K$-theory, Publ. Math. IHES 34, 129-151, 1968. MR 0234452 (38:2769)

[Sol67]
L. Solomon, The Burnside algebra of a finite group, J. Combin. Theory 2, 603-615, 1967. MR 0214679 (35:5528)

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 55P91, 19A22, 55P42

Retrieve articles in all Journals with MSC (2000): 55P91, 19A22, 55P42

Additional Information:

Halvard Fausk
Affiliation: Department of Mathematics, University of Oslo, 1053 Blindern, 0316 Oslo, Norway
Email: fausk@math.uio.no

DOI: 10.1090/S0002-9947-08-04528-5
PII: S 0002-9947(08)04528-5
Received by editor(s): December 18, 2006
Posted: April 14, 2008
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google