The Laitinen Conjecture for finite solvable Oliver groups
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- by Krzysztof Pawałowski and Toshio Sumi PDF
- Proc. Amer. Math. Soc. 137 (2009), 2147-2156 Request permission
Abstract:
For smooth actions of $G$ on spheres with exactly two fixed points, the Laitinen Conjecture proposed an answer to the Smith question about the $G$-modules determined on the tangent spaces at the two fixed points. Morimoto obtained the first counterexample to the Laitinen Conjecture for $G = \textrm {Aut}(A_6)$. By answering the Smith question for some finite solvable Oliver groups $G$, we obtain new counterexamples to the Laitinen Conjecture, presented for the first time in the case where $G$ is solvable.References
- M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. II. Applications, Ann. of Math. (2) 88 (1968), 451–491. MR 232406, DOI 10.2307/1970721
- Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
- 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
- Ju, X.M., The Smith set of the group $S_5 \times C_2 \times \dots \times C_2$, accepted for publication in Osaka Journal of Mathematics.
- Katsuo Kawakubo, The theory of transformation groups, Translated from the 1987 Japanese edition, The Clarendon Press, Oxford University Press, New York, 1991. MR 1150492
- Koto, A., Morimoto, M., and Qi, Y., The Smith sets of finite groups with normal Sylow $2$-subgroups and small nilquotients, J. Math. Kyoto Univ. 48 (2008), 219–227.
- Erkki Laitinen and Masaharu Morimoto, Finite groups with smooth one fixed point actions on spheres, Forum Math. 10 (1998), no. 4, 479–520. MR 1631012, DOI 10.1515/form.10.4.479
- Erkki Laitinen and Krzysztof Pawałowski, Smith equivalence of representations for finite perfect groups, Proc. Amer. Math. Soc. 127 (1999), no. 1, 297–307. MR 1468195, DOI 10.1090/S0002-9939-99-04544-X
- Morimoto, M., Smith equivalent Aut$(A_6)$-representations are isomorphic, Proc. Amer. Math. Soc. 136 (2008), 3683–3688.
- Masaharu Morimoto, Toshio Sumi, and Mamoru Yanagihara, Finite groups possessing gap modules, Geometry and topology: Aarhus (1998), Contemp. Math., vol. 258, Amer. Math. Soc., Providence, RI, 2000, pp. 329–342. MR 1778115, DOI 10.1090/conm/258/1778115
- Robert Oliver, Fixed-point sets of group actions on finite acyclic complexes, Comment. Math. Helv. 50 (1975), 155–177. MR 375361, DOI 10.1007/BF02565743
- Krzysztof Pawałowski, Smith equivalence of group modules and the Laitinen conjecture. A survey, Geometry and topology: Aarhus (1998), Contemp. Math., vol. 258, Amer. Math. Soc., Providence, RI, 2000, pp. 343–350. MR 1778116, DOI 10.1090/conm/258/1778116
- Krzysztof Pawałowski and Ronald Solomon, Smith equivalence and finite Oliver groups with Laitinen number 0 or 1, Algebr. Geom. Topol. 2 (2002), 843–895. MR 1936973, DOI 10.2140/agt.2002.2.843
- Krzysztof Pawałowski and Toshio Sumi, Finite groups with Smith equivalent, nonisomorphic representations, Proceedings of 34th Symposium on Transformation Groups, Wing Co., Wakayama, 2007, pp. 68–76. MR 2313386
- Cristian U. Sanchez, Actions of groups of odd order on compact, orientable manifolds, Proc. Amer. Math. Soc. 54 (1976), 445–448. MR 407871, DOI 10.1090/S0002-9939-1976-0407871-X
- Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 0450380, DOI 10.1007/978-1-4684-9458-7
- P. A. Smith, New results and old problems in finite transformation groups, Bull. Amer. Math. Soc. 66 (1960), 401–415. MR 125581, DOI 10.1090/S0002-9904-1960-10491-0
- Toshio Sumi, Gap modules for direct product groups, J. Math. Soc. Japan 53 (2001), no. 4, 975–990. MR 1852892, DOI 10.2969/jmsj/05340975
- Toshio Sumi, Gap modules for semidirect product groups, Kyushu J. Math. 58 (2004), no. 1, 33–58. MR 2053718, DOI 10.2206/kyushujm.58.33
- Sumi, T., Finite groups possessing Smith equivalent, nonisomorphic representations, RIMS Kokyuroku no. 1569, Kyoto Univ. (2007), 170–179.
- The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4, 2006 http://www.gap-system.org.
Additional Information
- Krzysztof Pawałowski
- Affiliation: Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Umultowska 87, 61-614 Poznań, Poland
- Email: kpa@amu.edu.pl
- Toshio Sumi
- Affiliation: Department of Art and Information Design, Faculty of Design, Kyushu University, 4-9-1 Shiobaru, Minami-ku, Fukuoka, 815-8540, Japan
- Email: sumi@design.kyushu-u.ac.jp
- Received by editor(s): May 2, 2008
- Received by editor(s) in revised form: August 4, 2008
- Published electronically: January 26, 2009
- Communicated by: Paul Goerss
- © Copyright 2009 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 137 (2009), 2147-2156
- MSC (2000): Primary 57S17, 57S25
- DOI: https://doi.org/10.1090/S0002-9939-09-09719-6
- MathSciNet review: 2480297