Papers
Topics
Authors
Recent
Search
2000 character limit reached

Free torus actions and twisted suspensions

Published 10 May 2023 in math.GT and math.DG | (2305.06068v3)

Abstract: We express the total space of a principal circle bundle over a connected sum of two manifolds in terms of the total spaces of circle bundles over each summand, provided certain conditions hold. We then apply this result to provide sufficient conditions for the existence of free circle and torus actions on connected sums of products of spheres and obtain a topological classification of closed, simply-connected manifolds with a free cohomogeneity-four torus action. As a corollary, we obtain infinitely-many manifolds with Riemannian metrics of positive Ricci curvature and isometric torus actions.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (53)
  1. Emil Artin. Zur Isotopie zweidimensionaler Flächen im R4subscript𝑅4R_{4}italic_R start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT. Abh. Math. Sem. Univ. Hamburg, 4(1):174–177, 1925. doi:10.1007/BF02950724.
  2. D. Barden. Simply connected five-manifolds. Ann. of Math. (2), 82:365–385, 1965. doi:10.2307/1970702.
  3. Glen E. Bredon. Introduction to compact transformation groups. Pure and Applied Mathematics, Vol. 46. Academic Press, New York-London, 1972.
  4. Bradley Lewis Burdick. Ricci-positive metrics on connected sums of projective spaces. Differential Geom. Appl., 62:212–233, 2019. doi:10.1016/j.difgeo.2018.11.005.
  5. Bradley Lewis Burdick. Metrics of positive Ricci curvature on the connected sums of products with arbitrarily many spheres. Ann. Global Anal. Geom., 58(4):433–476, 2020. doi:10.1007/s10455-020-09732-7.
  6. Almost free actions on manifolds. Bull. Austral. Math. Soc., 10:177–196, 1974. doi:10.1017/S000497270004082X.
  7. Positive Ricci curvature on simply-connected manifolds with cohomogeneity-two torus actions. Proc. Amer. Math. Soc., 148(7):3087–3097, 2020. doi:10.1090/proc/14961.
  8. Lecture notes in algebraic topology, volume 35 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001. doi:10.1090/gsm/035.
  9. Nonconnected moduli spaces of nonnegative sectional curvature metrics on simply connected manifolds. Bull. Lond. Math. Soc., 50(1):96–107, 2018. doi:10.1112/blms.12095.
  10. Haibao Duan. Circle actions and suspension operations on smooth manifolds, 2022. arXiv:2202.06225.
  11. Circle bundles over 4-manifolds. Arch. Math. (Basel), 85(3):278–282, 2005. doi:10.1007/s00013-005-1214-4.
  12. Allan L. Edmonds. A survey of group actions on 4-manifolds. In Handbook of group actions. Vol. III, volume 40 of Adv. Lect. Math. (ALM), pages 421–460. Int. Press, Somerville, MA, 2018.
  13. Algebraic models in geometry, volume 17 of Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, 2008.
  14. Torus actions on rationally elliptic manifolds. Math. Z., 297(1-2):197–221, 2021. doi:10.1007/s00209-020-02508-6.
  15. Cohomogeneity-two torus actions on non-negatively curved manifolds of low dimension. Math. Z., 276(1-2):133–152, 2014. doi:10.1007/s00209-013-1190-5.
  16. Invariant metrics of positive Ricci curvature on principal bundles. Math. Z., 227(3):455–463, 1998. doi:10.1007/PL00004385.
  17. A classification of 6666-manifolds with free S1superscript𝑆1S^{1}italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT actions. In Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971), Part I, Lecture Notes in Math., Vol. 298, pages 316–323. Springer, Berlin, 1972.
  18. McFeely Jackson Goodman. Moduli spaces of Ricci positive metrics in dimension five. Geom. Topol., in press. arXiv:2002.00333.
  19. Karsten Grove. Geometry of, and via, symmetries. In Conformal, Riemannian and Lagrangian geometry (Knoxville, TN, 2000), volume 27 of Univ. Lecture Ser., pages 31–53. Amer. Math. Soc., Providence, RI, 2002. doi:10.1090/ulect/027/02.
  20. Semi-free actions with manifold orbit spaces. Doc. Math., 25:2085–2114, 2020.
  21. Lifting compact group actions in fiber bundles. Japan. J. Math. (N.S.), 2(1):13–25, 1976. doi:10.4099/math1924.2.13.
  22. Morris W. Hirsch. Differential topology. Graduate Texts in Mathematics, No. 33. Springer-Verlag, New York-Heidelberg, 1976.
  23. Dale Husemoller. Fibre bundles, volume 20 of Graduate Texts in Mathematics. Springer-Verlag, New York, third edition, 1994. doi:10.1007/978-1-4757-2261-1.
  24. Yi Jiang. Regular circle actions on 2-connected 7-manifolds. J. Lond. Math. Soc. (2), 90(2):373–387, 2014. doi:10.1112/jlms/jdu028.
  25. Torus group actions on simply connected manifolds. Pacific J. Math., 53:435–444, 1974. URL: http://projecteuclid.org/euclid.pjm/1102911611.
  26. Shoshichi Kobayashi. Fixed points of isometries. Nagoya Math. J., 13:63–68, 1958. URL: http://projecteuclid.org/euclid.nmj/1118800030.
  27. János Kollár. Circle actions on simply connected 5-manifolds. Topology, 45(3):643–671, 2006. doi:10.1016/j.top.2006.01.003.
  28. Antoni A. Kosinski. Differential manifolds, volume 138 of Pure and Applied Mathematics. Academic Press, Inc., Boston, MA, 1993.
  29. Nonconnected moduli spaces of positive sectional curvature metrics. J. Amer. Math. Soc., 6(4):825–850, 1993. doi:10.2307/2152742.
  30. H. Blaine Lawson, Jr. and Marie-Louise Michelsohn. Spin geometry, volume 38 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1989.
  31. J. Levine. Semi-free circle actions on spheres. Invent. Math., 22:161–186, 1973. doi:10.1007/BF01392300.
  32. Characteristic classes. Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 76.
  33. John C. Nash. Positive Ricci curvature on fibre bundles. J. Differential Geometry, 14(2):241–254, 1979. URL: http://projecteuclid.org/euclid.jdg/1214434973.
  34. Hae Soo Oh. 6666-dimensional manifolds with effective T4superscript𝑇4T^{4}italic_T start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT-actions. Topology Appl., 13(2):137–154, 1982. doi:10.1016/0166-8641(82)90016-5.
  35. Hae Soo Oh. Toral actions on 5555-manifolds. Trans. Amer. Math. Soc., 278(1):233–252, 1983. doi:10.2307/1999313.
  36. Actions of SO⁢(2)SO2{\rm SO}(2)roman_SO ( 2 ) on 3-manifolds. In Proc. Conf. on Transformation Groups (New Orleans, La., 1967), pages 297–318. Springer, New York, 1968.
  37. Actions of the torus on 4444-manifolds. I. Trans. Amer. Math. Soc., 152:531–559, 1970. doi:10.2307/1995586.
  38. Richard S. Palais. Natural operations on differential forms. Trans. Amer. Math. Soc., 92:125–141, 1959. doi:10.2307/1993171.
  39. On ℤ2subscriptℤ2\mathbb{Z}_{2}blackboard_Z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and 𝕊1superscript𝕊1\mathbb{S}^{1}blackboard_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT free actions on spaces of cohomology type (a,b)𝑎𝑏(a,b)( italic_a , italic_b ). Houston J. Math., 36(1):137–146, 2010. doi:10.1002/nag.993.
  40. Frank Raymond. Classification of the actions of the circle on 3333-manifolds. Trans. Amer. Math. Soc., 131:51–78, 1968. doi:10.2307/1994680.
  41. Philipp Reiser. Metrics of positive Ricci curvature on simply-connected manifolds of dimension 6⁢k6𝑘6k6 italic_k, 2022. arXiv:2210.15610.
  42. Philipp Reiser. Generalized surgery on Riemannian manifolds of positive Ricci curvature. Trans. Amer. Math. Soc., 376(5):3397–3418, 2023. doi:10.1090/tran/8789.
  43. Catherine Searle. Symmetries of spaces with lower curvature bounds. Notices Amer. Math. Soc., 70(4):564–575, 2023. doi:10.1090/noti2651.
  44. Positive Ricci curvature on the connected sums of Sn×Smsuperscript𝑆𝑛superscript𝑆𝑚S^{n}\times S^{m}italic_S start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT × italic_S start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT. J. Differential Geom., 33(1):127–137, 1991. URL: http://projecteuclid.org/euclid.jdg/1214446032.
  45. Stephen Smale. On the structure of 5555-manifolds. Ann. of Math. (2), 75:38–46, 1962. doi:10.2307/1970417.
  46. Norman Steenrod. The topology of fibre bundles. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1999. Reprint of the 1957 edition, Princeton Paperbacks.
  47. J. C. Su. Transformation groups on cohomology projective spaces. Trans. Amer. Math. Soc., 106:305–318, 1963. doi:10.2307/1993772.
  48. Alexander I. Suciu. Iterated spinning and homology spheres. Trans. Amer. Math. Soc., 321(1):145–157, 1990. doi:10.2307/2001595.
  49. Moduli spaces of Riemannian metrics, volume 46 of Oberwolfach Seminars. Birkhäuser Verlag, Basel, 2015. Second corrected printing. doi:10.1007/978-3-0348-0948-1.
  50. C. T. C. Wall. Diffeomorphisms of 4444-manifolds. J. London Math. Soc., 39:131–140, 1964. doi:10.1112/jlms/s1-39.1.131.
  51. Einstein metrics on principal torus bundles. J. Differential Geom., 31(1):215–248, 1990. URL: http://projecteuclid.org/euclid.jdg/1214444095.
  52. Hassler Whitney. Differentiable manifolds. Ann. of Math. (2), 37(3):645–680, 1936. doi:10.2307/1968482.
  53. Burkhard Wilking. Nonnegatively and positively curved manifolds. In Surveys in differential geometry. Vol. XI, volume 11 of Surv. Differ. Geom., pages 25–62. Int. Press, Somerville, MA, 2007. doi:10.4310/SDG.2006.v11.n1.a3.
Citations (4)
View on Semantic Scholar

Summary

No one has generated a summary of this paper yet.

Sign Up to Summarize

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Generate Now

Collections

Sign up for free to add this paper to one or more collections.

Sign Up

Tweets

Sign up for free to view the 1 tweet with 0 likes about this paper.

Sign Up for Free