Compact group actions and maps into 𝐾(𝜋,1)-spaces

Gottlieb, Daniel H.; Lee, Kyung B.; Özaydin, Murad

doi:10.1090/S0002-9947-1985-0766228-2

Compact group actions and maps into $K(\pi ,1)$-spaces
HTML articles powered by AMS MathViewer

by Daniel H. Gottlieb, Kyung B. Lee and Murad Özaydin PDF

Trans. Amer. Math. Soc. 287 (1985), 419-429 Request permission

Abstract:

Let $G$ act on an aspherical manifold $M$. If $G$ is a compact Lie group acting effectively and homotopically trivially then $G$ must be abelian. We prove a much more general form of this result, thus extending results of Donnelly and Schultz. Our method gives us a splitting result for torus actions complementing a result of Conner and Raymond. We also generalize a theorem of Schoen and Yau on homotopy equivariance.

References

M. A. Armstrong, The fundamental group of the orbit space of a discontinuous group, Proc. Cambridge Philos. Soc. 64 (1968), 299–301. MR 221488, DOI 10.1017/s0305004100042845
M. A. Armstrong, Calculating the fundamental group of an orbit space, Proc. Amer. Math. Soc. 84 (1982), no. 2, 267–271. MR 637181, DOI 10.1090/S0002-9939-1982-0637181-X
Armand Borel, Topics in the homology theory of fibre bundles, Lecture Notes in Mathematics, No. 36, Springer-Verlag, Berlin-New York, 1967. Lectures given at the University of Chicago, 1954; Notes by Edward Halpern. MR 0221507, DOI 10.1007/BFb0096867
Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
P. E. Conner and Frank Raymond, Actions of compact Lie groups on aspherical manifolds, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969) Markham, Chicago, Ill., 1970, pp. 227–264. MR 0271958
P. E. Conner and Frank Raymond, Injective operations of the toral groups, Topology 10 (1971), 283–296. MR 281218, DOI 10.1016/0040-9383(71)90021-8
P. E. Conner and Frank Raymond, Manifolds with few periodic homeomorphisms, Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971) Lecture Notes in Math., Vol. 299, Springer, Berlin, 1972, pp. 1–75. MR 0358835
Harold Donnelly and Reinhard Schultz, Compact group actions and maps into aspherical manifolds, Topology 21 (1982), no. 4, 443–455. MR 670746, DOI 10.1016/0040-9383(82)90022-2
Daniel Henry Gottlieb, The Lefschetz number and Borsuk-Ulam theorems, Pacific J. Math. 103 (1982), no. 1, 29–37. MR 687962, DOI 10.2140/pjm.1982.103.29
Steven P. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2) 117 (1983), no. 2, 235–265. MR 690845, DOI 10.2307/2007076
K. B. Lee and Frank Raymond, Topological, affine and isometric actions on flat Riemannian manifolds, J. Differential Geometry 16 (1981), no. 2, 255–269. MR 638791
Kyung B. Lee and Frank Raymond, Topological, affine and isometric actions on flat Riemannian manifolds. II, Topology Appl. 13 (1982), no. 3, 295–310. MR 651511, DOI 10.1016/0166-8641(82)90037-2
Kyung Bai Lee and Frank Raymond, Geometric realization of group extensions by the Seifert construction, Contributions to group theory, Contemp. Math., vol. 33, Amer. Math. Soc., Providence, RI, 1984, pp. 353–411. MR 767121, DOI 10.1090/conm/033/767121
Saunders Mac Lane, Homology, Die Grundlehren der mathematischen Wissenschaften, Band 114, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0156879, DOI 10.1007/978-3-642-62029-4
F. Rhodes, On the fundamental group of a transformation group, Proc. London Math. Soc. (3) 16 (1966), 635–650. MR 203715, DOI 10.1112/plms/s3-16.1.635
Richard Schoen and Shing Tung Yau, Compact group actions and the topology of manifolds with nonpositive curvature, Topology 18 (1979), no. 4, 361–380. MR 551017, DOI 10.1016/0040-9383(79)90026-0
Reinhard Schultz, Group actions on hypertoral manifolds. I, Topology Symposium, Siegen 1979 (Proc. Sympos., Univ. Siegen, Siegen, 1979), Lecture Notes in Math., vol. 788, Springer, Berlin, 1980, pp. 364–377. MR 585669
Heiner Zieschang, Elmar Vogt, and Hans-Dieter Coldewey, Surfaces and planar discontinuous groups, Lecture Notes in Mathematics, vol. 835, Springer, Berlin, 1980. Translated from the German by John Stillwell. MR 606743, DOI 10.1007/BFb0089692