Proper actions on cohomology manifolds

Biller, Harald

doi:10.1090/S0002-9947-02-03123-9

Proper actions on cohomology manifolds
HTML articles powered by AMS MathViewer

by Harald Biller PDF

Trans. Amer. Math. Soc. 355 (2003), 407-432 Request permission

Abstract:

Essential results about actions of compact Lie groups on connected manifolds are generalized to proper actions of arbitrary groups on connected cohomology manifolds. Slices are replaced by certain fiber bundle structures on orbit neighborhoods. The group dimension is shown to be effectively finite. The orbits of maximal dimension form a dense open connected subset. If some orbit has codimension at most $2$, then the group is effectively a Lie group.

References

Herbert Abels, Parallelizability of proper actions, global $K$-slices and maximal compact subgroups, Math. Ann. 212 (1974/75), 1–19. MR 375264, DOI 10.1007/BF01343976
C. Allday and V. Puppe, Cohomological methods in transformation groups, Cambridge Studies in Advanced Mathematics, vol. 32, Cambridge University Press, Cambridge, 1993. MR 1236839, DOI 10.1017/CBO9780511526275
Paul Baum, Alain Connes, and Nigel Higson, Classifying space for proper actions and $K$-theory of group $C^\ast$-algebras, $C^\ast$-algebras: 1943–1993 (San Antonio, TX, 1993) Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240–291. MR 1292018, DOI 10.1090/conm/167/1292018
Harald Biller, Actions of compact groups on spheres and on generalized quadrangles, Dissertation, Stuttgart, 1999, http://elib.uni-stuttgart.de/opus/volltexte/1999/566.
Harald Biller, Actions of compact groups on compact $(4,m)$-quadrangles, Geom. Dedicata 83 (2000), no. 1-3, 273–312. Special issue dedicated to Helmut R. Salzmann on the occasion of his 70th birthday. MR 1800024, DOI 10.1023/A:1005292209317
—, Fixed points of pro-tori in cohomology spheres, Preprint 2183, Fachbereich Mathematik, Technische Universität Darmstadt, 2001, http://wwwbib.mathematik.tu-darmstadt. de/Math-Net/Preprints/Listen/shadow/pp2183.html.
—, Characterizations of proper actions, Preprint 2211, Fachbereich Mathematik, Technische Universität Darmstadt, 2002, http://wwwbib.mathematik.tu-darmstadt.de/ Math-Net/Preprints/Listen/shadow/pp2211.html; to appear in Math. Proc. Cambridge Philosophical Society.
A. R. Collar, On the reciprocation of certain matrices, Proc. Roy. Soc. Edinburgh 59 (1939), 195–206. MR 8
Armand Borel, Seminar on transformation groups, Annals of Mathematics Studies, No. 46, Princeton University Press, Princeton, N.J., 1960. With contributions by G. Bredon, E. E. Floyd, D. Montgomery, R. Palais. MR 0116341
N. Bourbaki, Éléments de mathématique. Topologie générale. Chapitres 1 à 4, Hermann, Paris, 1971. MR 0358652
Glen E. Bredon, Some theorems on transformation groups, Ann. of Math. (2) 67 (1958), 104–118. MR 93559, DOI 10.2307/1969930
Glen E. Bredon, Orientation in generalized manifolds and applications to the theory of transformation groups, Michigan Math. J. 7 (1960), 35–64. MR 116342
Glen E. Bredon, Cohomology fibre spaces, the Smith-Gysin sequence, and orientation in generalized manifolds, Michigan Math. J. 10 (1963), 321–333. MR 159329
Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
Glen E. Bredon, Topology and geometry, Graduate Texts in Mathematics, vol. 139, Springer-Verlag, New York, 1993. MR 1224675, DOI 10.1007/978-1-4757-6848-0
Glen E. Bredon, Sheaf theory, 2nd ed., Graduate Texts in Mathematics, vol. 170, Springer-Verlag, New York, 1997. MR 1481706, DOI 10.1007/978-1-4612-0647-7
G. E. Bredon, Frank Raymond, and R. F. Williams, $p$-adic groups of transformations, Trans. Amer. Math. Soc. 99 (1961), 488–498. MR 142682, DOI 10.1090/S0002-9947-1961-0142682-8
J. Bryant, S. Ferry, W. Mio, and S. Weinberger, Topology of homology manifolds, Ann. of Math. (2) 143 (1996), no. 3, 435–467. MR 1394965, DOI 10.2307/2118532
F. Buekenhout (ed.), Handbook of incidence geometry, North-Holland, Amsterdam, 1995. Buildings and foundations. MR 1360715
Tammo tom Dieck, Transformation groups, De Gruyter Studies in Mathematics, vol. 8, Walter de Gruyter & Co., Berlin, 1987. MR 889050, DOI 10.1515/9783110858372.312
James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
Ryszard Engelking, General topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR 1039321
Ryszard Engelking, Teoria wymiaru, Biblioteka Matematyczna, Tom 51. [Mathematics Library, Vol. 51], Państwowe Wydawnictwo Naukowe, Warsaw, 1977 (Polish). MR 0482696
P. Erdös, Note on products of consecutive integers, J. London Math. Soc. 14 (1939), 194–198. MR 22, DOI 10.1112/jlms/s1-14.3.194
A. Bouziad, M. Lemańczyk, and M. K. Mentzen, A compact monothetic semitopological semigroup whose set of idempotents is not closed, Semigroup Forum 62 (2001), no. 1, 98–102. MR 1832257, DOI 10.1007/s002330010016
Karl H. Hofmann and Sidney A. Morris, The structure of compact groups, De Gruyter Studies in Mathematics, vol. 25, Walter de Gruyter & Co., Berlin, 1998. A primer for the student—a handbook for the expert. MR 1646190
Karl H. Hofmann and Christian Terp, Compact subgroups of Lie groups and locally compact groups, Proc. Amer. Math. Soc. 120 (1994), no. 2, 623–634. MR 1166357, DOI 10.1090/S0002-9939-1994-1166357-9
Wu-yi Hsiang, Cohomology theory of topological transformation groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 85, Springer-Verlag, New York-Heidelberg, 1975. MR 0423384
Irving Kaplansky, Lie algebras and locally compact groups, University of Chicago Press, Chicago, Ill.-London, 1971. MR 0276398
Linus Kramer, Compact polygons, Dissertation, Tübingen, 1994.
Rainer Löwen, Locally compact connected groups acting on Euclidean space with Lie isotropy groups are Lie, Geometriae Dedicata 5 (1976), no. 2, 171–174. MR 419673, DOI 10.1007/BF00145954
Rainer Löwen, Topology and dimension of stable planes: on a conjecture of H. Freudenthal, J. Reine Angew. Math. 343 (1983), 108–122. MR 705880, DOI 10.1515/crll.1983.343.108
Gaven J. Martin, The Hilbert-Smith conjecture for quasiconformal actions, Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 66–70. MR 1694197, DOI 10.1090/S1079-6762-99-00062-1
P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
Deane Montgomery and Leo Zippin, Topological transformation groups I, Ann. of. Math., II. Ser. 41 (1940), 778–791.
J. J. Corliss, Upper limits to the real roots of a real algebraic equation, Amer. Math. Monthly 46 (1939), 334–338. MR 4
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Richard S. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. of Math. (2) 73 (1961), 295–323. MR 126506, DOI 10.2307/1970335
B. A. Pasynkov, On the coincidence of various definitions of dimensionality for factor spaces of locally bicompact groups, Uspehi Mat. Nauk 17 (1962), no. 5 (107), 129–135 (Russian). MR 0144998
Frank Raymond, The orbit spaces of totally disconnected groups of transformations on manifolds, Proc. Amer. Math. Soc. 12 (1961), 1–7. MR 141730, DOI 10.1090/S0002-9939-1961-0141730-4
Frank Raymond, Some remarks on the coefficients used in the theory of homology manifolds, Pacific J. Math. 15 (1965), 1365–1376. MR 189021
Du an Repov and Evgenij epin, A proof of the Hilbert-Smith conjecture for actions by Lipschitz maps, Math. Ann. 308 (1997), no. 2, 361–364. MR 1464908, DOI 10.1007/s002080050080
Helmut Salzmann, Dieter Betten, Theo Grundhöfer, Hermann Hähl, Rainer Löwen, and Markus Stroppel, Compact projective planes, De Gruyter Expositions in Mathematics, vol. 21, Walter de Gruyter & Co., Berlin, 1995. With an introduction to octonion geometry. MR 1384300, DOI 10.1515/9783110876833
E. G. Skljarenko, On the topological structure of locally bicompact groups and their factor spaces, Mat. Sb. (N.S.) 60 (102) (1963), 63–88 (Russian). MR 0147575
Bernhild Stroppel and Markus Stroppel, The automorphism group of a compact generalized quadrangle has finite dimension, Arch. Math. (Basel) 66 (1996), no. 1, 77–79. MR 1363780, DOI 10.1007/BF01323985
Bernhild Stroppel and Markus Stroppel, The automorphism group of a compact generalized polygon has finite dimension, Monatsh. Math. 127 (1999), no. 4, 343–347. MR 1694822, DOI 10.1007/s006050050046
Harald Upmeier, Symmetric Banach manifolds and Jordan $C^\ast$-algebras, North-Holland Mathematics Studies, vol. 104, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 96. MR 776786
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
C. T. Yang, Transformation groups on a homological manifold, Trans. Amer. Math. Soc. 87 (1958), 261–283. MR 100271, DOI 10.1090/S0002-9947-1958-0100271-5
Chung-Tao Yang, $p$-adic transformation groups, Michigan Math. J. 7 (1960), 201–218. MR 120310