Smith equivalent ${\pmb {\textrm {{Aut}}(A_6)}}$-representations are isomorphic
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Abstract:
Many authors, e.g. M. Atiyah, R. Bott, J. Milnor, G. Bredon, S. Cappell, J. Shaneson, C. Sanchez, T. Petrie, E. Laitinen, K. Pawałowski, R. Solomon and so on, studied Smith equivalent representations for finite groups. Observing their results, E. Laitinen conjectured that nonisomorphic Smith equivalent real $G$-modules exist if $a_G$, the number of real conjugacy classes of elements not of prime power order in $G$, is greater than or equal to $2$. This paper shows that in the case $G = \textrm {{Aut}}(A_6)$, $a_G = 2$ any two Smith equivalent real $G$-modules are isomorphic.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, Representations at fixed points of smooth actions of compact groups, Ann. of Math. (2) 89 (1969), 515–532. MR 246324, DOI 10.2307/1970649
- Sylvain E. Cappell and Julius L. Shaneson, Fixed points of periodic maps, Proc. Nat. Acad. Sci. U.S.A. 77 (1980), no. 9, 5052–5054. MR 587279, DOI 10.1073/pnas.77.9.5052
- Sylvain E. Cappell and Julius L. Shaneson, Fixed points of periodic differentiable maps, Invent. Math. 68 (1982), no. 1, 1–19. MR 666635, DOI 10.1007/BF01394267
- Sylvain E. Cappell and Julius L. Shaneson, Representations at fixed points, Group actions on manifolds (Boulder, Colo., 1983) Contemp. Math., vol. 36, Amer. Math. Soc., Providence, RI, 1985, pp. 151–158. MR 780963, DOI 10.1090/conm/036/780963
- Eung Chun Cho and Dong Youp Suh, Induction in equivariant $K$-theory and $s$-Smith equivalence of representations, Group actions on manifolds (Boulder, Colo., 1983) Contemp. Math., vol. 36, Amer. Math. Soc., Providence, RI, 1985, pp. 311–315. MR 780969, DOI 10.1090/conm/036/780969
- Eung Chun Cho, $s$-Smith equivalent representations of dihedral groups, Pacific J. Math. 135 (1988), no. 1, 17–28. MR 965682
- P. E. Conner and E. E. Floyd, Differentiable periodic maps, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 33, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1964. MR 0176478
- Karl Heinz Dovermann and Ted Petrie, Smith equivalence of representations for odd order cyclic groups, Topology 24 (1985), no. 3, 283–305. MR 815481, DOI 10.1016/0040-9383(85)90003-5
- Karl Heinz Dovermann, Ted Petrie, and Reinhard Schultz, Transformation groups and fixed point data, Group actions on manifolds (Boulder, Colo., 1983) Contemp. Math., vol. 36, Amer. Math. Soc., Providence, RI, 1985, pp. 159–189. MR 780964, DOI 10.1090/conm/036/780964
- Karl Heinz Dovermann and Dong Youp Suh, Smith equivalence for finite abelian groups, Pacific J. Math. 152 (1992), no. 1, 41–78. MR 1139972
- Karl Heinz Dovermann and Lawrence C. Washington, Relations between cyclotomic units and Smith equivalence of representations, Topology 28 (1989), no. 1, 81–89. MR 991100, DOI 10.1016/0040-9383(89)90033-5
- GAP, Groups, Algorithms, Programming, a System for Computational Discrete Algebra, Release 4.3, 06 May 2002, http://www.gap-system.org.
- 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
- Mikiya Masuda and Ted Petrie, Lectures on transformation groups and Smith equivalence, Group actions on manifolds (Boulder, Colo., 1983) Contemp. Math., vol. 36, Amer. Math. Soc., Providence, RI, 1985, pp. 191–242. MR 780965, DOI 10.1090/conm/036/780965
- J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426. MR 196736, DOI 10.1090/S0002-9904-1966-11484-2
- 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
- Robert Oliver, Smooth compact Lie group actions on disks, Math. Z. 149 (1976), no. 1, 79–96. MR 423390, DOI 10.1007/BF01301634
- 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
- Ted Petrie, $G$ surgery. I. A survey, Algebraic and geometric topology (Proc. Sympos., Univ. California, Santa Barbara, Calif., 1977) Lecture Notes in Math., vol. 664, Springer, Berlin, 1978, pp. 197–233. MR 518416
- 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
- Ted Petrie, Three theorems in transformation groups, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), Lecture Notes in Math., vol. 763, Springer, Berlin, 1979, pp. 549–572. MR 561238
- Ted Petrie, The equivariant $J$ homomorphism and Smith equivalence of representations, Current trends in algebraic topology, Part 2 (London, Ont., 1981) CMS Conf. Proc., vol. 2, Amer. Math. Soc., Providence, R.I., 1982, pp. 223–233. MR 686148
- Ted Petrie, Smith equivalence of representations, Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 1, 61–99. MR 704803, DOI 10.1017/S0305004100060928
- Ted Petrie and John D. Randall, Transformation groups on manifolds, Monographs and Textbooks in Pure and Applied Mathematics, vol. 82, Marcel Dekker, Inc., New York, 1984. MR 748850
- Ted Petrie and John Randall, Spherical isotropy representations, Inst. Hautes Études Sci. Publ. Math. 62 (1985), 221–256. MR 823175
- 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
- R. Schultz (ed.), Problems submitted to the AMS summer research conference on group actions, Group actions on manifolds (Boulder, Colo., 1983) Contemp. Math., vol. 36, Amer. Math. Soc., Providence, RI, 1985, pp. 513–568. MR 780979, DOI 10.1090/conm/036/780979
- 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
- Eung Chun Cho and Dong Youp Suh, Induction in equivariant $K$-theory and $s$-Smith equivalence of representations, Group actions on manifolds (Boulder, Colo., 1983) Contemp. Math., vol. 36, Amer. Math. Soc., Providence, RI, 1985, pp. 311–315. MR 780969, DOI 10.1090/conm/036/780969
Additional Information
- Masaharu Morimoto
- Affiliation: Graduate School of Natural Science and Technology, Okayama University, 3-1-1 Tsushimanaka, Okayama, 700-8530 Japan
- Email: morimoto@ems.okayama-u.ac.jp
- Received by editor(s): January 24, 2006
- Received by editor(s) in revised form: July 7, 2006
- Published electronically: June 3, 2008
- Additional Notes: The author was partially supported by Grant-in-Aid for Scientific Research (Kakenhi) 18540086
- Communicated by: Paul Goerss
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 3683-3688
- MSC (2000): Primary 57S17, 57S25, 55M35, 20C15
- DOI: https://doi.org/10.1090/S0002-9939-08-08891-6
- MathSciNet review: 2415055
Dedicated: Dedicated to Professor Katsuhiro Komiya on his 60th birthday