Seifert manifolds with fiber spherical space forms

Lee, Jong Bum; Lee, Kyung; Raymond, Frank

doi:10.1090/S0002-9947-96-01609-1

Seifert manifolds with fiber spherical space forms
HTML articles powered by AMS MathViewer

by Jong Bum Lee, Kyung Bai Lee and Frank Raymond PDF

Trans. Amer. Math. Soc. 348 (1996), 3763-3798 Request permission

Abstract:

We study the Seifert fiber spaces modeled on the product space $S^3 \times \mathbb {R}^2$. Such spaces are “fiber bundles” with singularities. The regular fibers are spherical space-forms of $S^3$, while singular fibers are finite quotients of regular fibers. For each of possible space-form groups $\Delta$ of $S^3$, we obtain a criterion for a group extension $\varPi$ of $\Delta$ to act on $S^3 \times \mathbb {R}^2$ as weakly $S^3$-equivariant maps, which gives rise to a Seifert fiber space modeled on $S^3 \times \mathbb {R}^2$ with weakly $S^3$-equivariant maps $\mathrm {TOP}_{S^3}(S^3 \times \mathbb {R}^2)$ as the universal group. In the course of proving our main results, we also obtain an explicit formula for $H^2(Q; \mathbb {Z})$ for a cocompact crystallographic or Fuchsian group $Q$. Most of our methods for $S^3$ apply to compact Lie groups with discrete center, and we state some of our results in this general context.

References

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, Holomorphic Seifert fiberings, Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971) Lecture Notes in Math., Vol. 299, Springer, Berlin, 1972, pp. 124–204. MR 0590802
P. E. Conner and Frank Raymond, Deforming homotopy equivalences to homeomorphisms in aspherical manifolds, Bull. Amer. Math. Soc. 83 (1977), no. 1, 36–85. MR 467777, DOI 10.1090/S0002-9904-1977-14179-7
Paul Igodt and Kyung Bai Lee, Application of group cohomology to space constructions, Trans. Amer. Math. Soc. 304 (1987), no. 1, 69–82. MR 906806, DOI 10.1090/S0002-9947-1987-0906806-2
Mark Jankins and Walter D. Neumann, Lectures on Seifert manifolds, Brandeis Lecture Notes, vol. 2, Brandeis University, Waltham, MA, 1983. MR 741334
Ravi S. Kulkarni and Frank Raymond, $3$-dimensional Lorentz space-forms and Seifert fiber spaces, J. Differential Geom. 21 (1985), no. 2, 231–268. MR 816671
K. B. Lee, Infra-solvmanifolds of type (R), Quart. J. Math. Oxford (2) 46 (1995), 185–195
Kyung Bai Lee and Frank Raymond, The role of Seifert fiber spaces in transformation groups, Group actions on manifolds (Boulder, Colo., 1983) Contemp. Math., vol. 36, Amer. Math. Soc., Providence, RI, 1985, pp. 367–425. MR 780974, DOI 10.1090/conm/036/780974
Kyung Bai Lee and Frank Raymond, Seifert manifolds modelled on principal bundles, Transformation groups (Osaka, 1987) Lecture Notes in Math., vol. 1375, Springer, Berlin, 1989, pp. 207–215. MR 1006694, DOI 10.1007/BFb0085611
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
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
L. Brink, P. di Vecchia, and P. Howe, A Lagrangian formulation of the classical and quantum dynamics of spinning particles, Nuclear Phys. B 118 (1977), no. 1-2, 76–94. MR 426651, DOI 10.1016/0550-3213(77)90364-9
M. M. Postnikov, Three-dimensional spherical forms, Trudy Mat. Inst. Steklov. 196 (1991), 114–146 (Russian). Translated in Proc. Steklov Inst. Math. 1992, no. 4, 129–161; Discrete geometry and topology (Russian). MR 1111299
Peter Scott, The geometries of $3$-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401–487. MR 705527, DOI 10.1112/blms/15.5.401
Frank Raymond and Alphonse T. Vasquez, $3$-manifolds whose universal coverings are Lie groups, Topology Appl. 12 (1981), no. 2, 161–179. MR 612013, DOI 10.1016/0166-8641(81)90018-3
Frank Raymond and David Wigner, Constructions of aspherical manifolds, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977) Lecture Notes in Math., vol. 657, Springer, Berlin, 1978, pp. 408–422. MR 513561
Joseph A. Wolf, Spaces of constant curvature, 3rd ed., Publish or Perish, Inc., Boston, Mass., 1974. MR 0343214