Isovariant Borsuk-Ulam results for pseudofree circle actions and their converse
HTML articles powered by AMS MathViewer
- by Ikumitsu Nagasaki PDF
- Trans. Amer. Math. Soc. 358 (2006), 743-757 Request permission
Abstract:
In this paper we shall study the existence of an $S^1$-isovariant map from a rational homology sphere $M$ with pseudofree action to a representation sphere $SW$. We first show some isovariant Borsuk-Ulam type results. Next we shall consider the converse of those results and show that there exists an $S^1$-isovariant map from $M$ to $SW$ under suitable conditions.References
- K. Borsuk, Drei Sätze über die $n$-dimensionale Sphäre, Fund. Math, 20 (1933), 177–190.
- Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
- William Browder and Frank Quinn, A surgery theory for $G$-manifolds and stratified sets, Manifolds—Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973) Univ. Tokyo Press, Tokyo, 1975, pp. 27–36. MR 0375348
- Mónica Clapp and Dieter Puppe, Critical point theory with symmetries, J. Reine Angew. Math. 418 (1991), 1–29. MR 1111200, DOI 10.1007/BF02566437
- 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
- Albrecht Dold, Simple proofs of some Borsuk-Ulam results, Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982) Contemp. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1983, pp. 65–69. MR 711043, DOI 10.1090/conm/019/711043
- Karl Heinz Dovermann, Almost isovariant normal maps, Amer. J. Math. 111 (1989), no. 6, 851–904. MR 1026286, DOI 10.2307/2374779
- Giora Dula and Reinhard Schultz, Diagram cohomology and isovariant homotopy theory, Mem. Amer. Math. Soc. 110 (1994), no. 527, viii+82. MR 1209409, DOI 10.1090/memo/0527
- Davide L. Ferrario, On the equivariant Hopf theorem, Topology 42 (2003), no. 2, 447–465. MR 1941444, DOI 10.1016/S0040-9383(02)00015-0
- M. Furuta, Monopole equation and the $\frac {11}8$-conjecture, Math. Res. Lett. 8 (2001), no. 3, 279–291. MR 1839478, DOI 10.4310/MRL.2001.v8.n3.a5
- Katsuo Kawakubo, The theory of transformation groups, Translated from the 1987 Japanese edition, The Clarendon Press, Oxford University Press, New York, 1991. MR 1150492
- Wolfgang Lück, Transformation groups and algebraic $K$-theory, Lecture Notes in Mathematics, vol. 1408, Springer-Verlag, Berlin, 1989. Mathematica Gottingensis. MR 1027600, DOI 10.1007/BFb0083681
- Jiří Matoušek, Using the Borsuk-Ulam theorem, Universitext, Springer-Verlag, Berlin, 2003. Lectures on topological methods in combinatorics and geometry; Written in cooperation with Anders Björner and Günter M. Ziegler. MR 1988723
- Deane Montgomery and C. T. Yang, Differentiable pseudo-free circle actions on homotopy seven spheres, Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971) Lecture Notes in Math., Vol. 298, Springer, Berlin, 1972, pp. 41–101. MR 0362383
- Ikumitsu Nagasaki, The weak isovariant Borsuk-Ulam theorem for compact Lie groups, Arch. Math. (Basel) 81 (2003), no. 3, 348–359. MR 2013267, DOI 10.1007/s00013-003-4693-1
- Ikumitsu Nagasaki, Isovariant maps between representation spaces, Sūrikaisekikenkyūsho K\B{o}kyūroku 1290 (2002), 83–94 (Japanese). Transformation groups from new points of view (Japanese) (Kyoto, 2002). MR 1982458
- Ted Petrie, Pseudoequivalences of $G$-manifolds, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 169–210. MR 520505
- H. Steinlein, Borsuk’s antipodal theorem and its generalizations and applications: a survey, Topological methods in nonlinear analysis, Sém. Math. Sup., vol. 95, Presses Univ. Montréal, Montreal, QC, 1985, pp. 166–235. MR 801938
- H. Steinlein, Spheres and symmetry: Borsuk’s antipodal theorem, Topol. Methods Nonlinear Anal. 1 (1993), no. 1, 15–33. MR 1215255, DOI 10.12775/TMNA.1993.004
- Arthur G. Wasserman, Isovariant maps and the Borsuk-Ulam theorem, Topology Appl. 38 (1991), no. 2, 155–161. MR 1094548, DOI 10.1016/0166-8641(91)90082-W
Additional Information
- Ikumitsu Nagasaki
- Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
- Email: nagasaki@math.sci.osaka-u.ac.jp
- Received by editor(s): March 1, 2004
- Published electronically: March 18, 2005
- Additional Notes: The author was partially supported by Grant-in-Aid for Scientific Research.
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 358 (2006), 743-757
- MSC (2000): Primary 55M20; Secondary 57S15, 55M25, 55S35
- DOI: https://doi.org/10.1090/S0002-9947-05-03822-5
- MathSciNet review: 2177039