Smith equivalent \pmb{𝐴𝑢𝑡(𝐴₆)}-representations are isomorphic

Morimoto, Masaharu

doi:10.1090/S0002-9939-08-08891-6

Smith equivalent ${\pmb {\textrm {{Aut}}(A_6)}}$-representations are isomorphic
HTML articles powered by AMS MathViewer

by Masaharu Morimoto PDF

Proc. Amer. Math. Soc. 136 (2008), 3683-3688 Request permission

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