Acyclic embeddings of open Riemann surfaces into new examples of elliptic manifolds
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- by Tyson Ritter
- Proc. Amer. Math. Soc. 141 (2013), 597-603
- DOI: https://doi.org/10.1090/S0002-9939-2012-11430-3
- Published electronically: June 21, 2012
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Abstract:
The geometric notion of ellipticity for complex manifolds was introduced by Gromov in his seminal 1989 paper on the Oka principle and is a sufficient condition for a manifold to be Oka. In the current paper we present contributions to three open questions involving elliptic and Oka manifolds. We show that quotients of $\mathbb {C}^n$ by discrete groups of affine transformations are elliptic. Combined with an example of Margulis, this yields new examples of elliptic manifolds with free fundamental groups and vanishing higher homotopy. Finally we show that every open Riemann surface embeds acyclically into an elliptic manifold, giving a partial answer to a question of Lárusson.References
- Gilbert Baumslag and James E. Roseblade, Subgroups of direct products of free groups, J. London Math. Soc. (2) 30 (1984), no. 1, 44–52. MR 760871, DOI 10.1112/jlms/s2-30.1.44
- James F. Davis and Paul Kirk, Lecture notes in algebraic topology, Graduate Studies in Mathematics, vol. 35, American Mathematical Society, Providence, RI, 2001. MR 1841974, DOI 10.1090/gsm/035
- Franc Forstnerič, Runge approximation on convex sets implies the Oka property, Ann. of Math. (2) 163 (2006), no. 2, 689–707. MR 2199229, DOI 10.4007/annals.2006.163.689
- Franc Forstnerič, Oka manifolds, C. R. Math. Acad. Sci. Paris 347 (2009), no. 17-18, 1017–1020 (English, with English and French summaries). MR 2554568, DOI 10.1016/j.crma.2009.07.005
- Franc Forstnerič and Finnur Lárusson, Survey of Oka theory, New York J. Math. 17A (2011), 11–38. MR 2782726
- David Fried and William M. Goldman, Three-dimensional affine crystallographic groups, Adv. in Math. 47 (1983), no. 1, 1–49. MR 689763, DOI 10.1016/0001-8708(83)90053-1
- Hans Grauert, Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen, Math. Ann. 133 (1957), 450–472. MR 0098198 (20:4660)
- Hans Grauert, On Levi’s problem and the imbedding of real-analytic manifolds, Ann. of Math. (2) 68 (1958), 460–472. MR 98847, DOI 10.2307/1970257
- M. Gromov, Oka’s principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2 (1989), no. 4, 851–897. MR 1001851, DOI 10.1090/S0894-0347-1989-1001851-9
- Finnur Lárusson, Excision for simplicial sheaves on the Stein site and Gromov’s Oka principle, Internat. J. Math. 14 (2003), no. 2, 191–209. MR 1966772, DOI 10.1142/S0129167X03001727
- Finnur Lárusson, Model structures and the Oka principle, J. Pure Appl. Algebra 192 (2004), no. 1-3, 203–223. MR 2067196, DOI 10.1016/j.jpaa.2004.02.005
- Finnur Lárusson, Mapping cylinders and the Oka principle, Indiana Univ. Math. J. 54 (2005), no. 4, 1145–1159. MR 2164421, DOI 10.1512/iumj.2005.54.2731
- G. A. Margulis, Free completely discontinuous groups of affine transformations, Dokl. Akad. Nauk SSSR 272 (1983), no. 4, 785–788 (Russian). MR 722330
- G. A. Margulis, Complete affine locally flat manifolds with a free fundamental group, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 134 (1984), 190–205 (Russian, with English summary). Automorphic functions and number theory, II. MR 741860
- John Milnor, On fundamental groups of complete affinely flat manifolds, Advances in Math. 25 (1977), no. 2, 178–187. MR 454886, DOI 10.1016/0001-8708(77)90004-4
- Kiyosi Oka, Sur les fonctions analytiques de plusieurs variables. III. Deuxième problème de Cousin, J. Sc. Hiroshima Univ. 9 (1939), 7–19.
- Tyson Ritter, A strong Oka principle for embeddings of some planar domains into $\mathbb C\times \mathbb C^*$, J. Geom. Anal. (to appear). arXiv:1011.4116
- Satoru Shimizu, Complex analytic properties of tubes over locally homogeneous hyperbolic affine manifolds, Tohoku Math. J. (2) 37 (1985), no. 3, 299–305. MR 799523, DOI 10.2748/tmj/1178228643
Bibliographic Information
- Tyson Ritter
- Affiliation: School of Mathematical Sciences, University of Adelaide, Adelaide SA 5005, Australia
- Email: tyson.ritter@adelaide.edu.au
- Received by editor(s): July 4, 2011
- Published electronically: June 21, 2012
- Communicated by: Franc Forstneric
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 597-603
- MSC (2010): Primary 32Q40; Secondary 32E10, 32H02, 32H35, 32M17, 32Q28
- DOI: https://doi.org/10.1090/S0002-9939-2012-11430-3
- MathSciNet review: 2996964