Hostname: page-component-cb9f654ff-hn9fh Total loading time: 0 Render date: 2025-08-31T18:35:16.488Z Has data issue: false hasContentIssue false
  • English
  • Français

The Laitinen Conjecture for finite non-solvable groups

Part of: Representation theory of groups Classical Topics in Algebraic Topology

Published online by Cambridge University Press:  05 December 2012

Krzysztof Pawałowski  and
Toshio Sumi
Krzysztof Pawałowski
Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Umultowska 87, 61-614 Poznań, Poland (kpa@amu.edu.pl)
Toshio Sumi
Affiliation:
Faculty of Arts and Science, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan (sumi@artsci.kyushu-u.ac.jp)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For any finite group G, we impose an algebraic condition, the Gnil-coset condition, and prove that any finite Oliver group G satisfying the Gnil-coset condition has a smooth action on some sphere with isolated fixed points at which the tangent G-modules are not isomorphic to each other. Moreover, we prove that, for any finite non-solvable group G not isomorphic to Aut(A6) or PΣL(2, 27), the Gnil-coset condition holds if and only if rG ≥ 2, where rG is the number of real conjugacy classes of elements of G not of prime power order. As a conclusion, the Laitinen Conjecture holds for any finite non-solvable group not isomorphic to Aut(A6).

Keywords

Smith equivalencereal G-moduleOliver groupLaitinen conjecturenon-solvable group

MSC classification

Secondary: 57S17: Finite transformation groups 57S25: Groups acting on specific manifolds 55M35: Finite groups of transformations (including Smith theory) 20C15: Ordinary representations and characters

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 1 , February 2013 , pp. 303 - 336
Copyright
Copyright © Edinburgh Mathematical Society 2012

References

1.Atiyah, M. F. and Bott, R., A Lefschetz fixed point formula for elliptic complexes, II, Applications, Annals Math. 88 (1968), 451–491.CrossRefGoogle Scholar
2.Bredon, G. E., Representations at fixed points of smooth actions of compact groups, Annals Math. (2) 89 (1969), 515–532.CrossRefGoogle Scholar
3.Bredon, G. E., Introduction to compact transformation groups, Pure and Applied Mathematics, Volume 6 (Academic Press, 1972).Google Scholar
4.Cappell, S. E. and Shaneson, J. L., Fixed points of periodic maps, Proc. Natl Acad. Sci. USA 77 (1980), 5052–5054.CrossRefGoogle ScholarPubMed
5.Cappell, S. E. and Shaneson, J. L., Fixed points of periodic differentiable maps, Invent. Math. 68 (1982), 1–19.CrossRefGoogle Scholar
6.Cappell, S. E. and Shaneson, J. L., Representations at fixed points, in Group actions on manifolds (ed. Schultz, R.), Contemporary Mathematics, Volume 36, pp. 151–158 (American Mathematical Society, Providence, RI, 1985).CrossRefGoogle Scholar
7.Cho, E. C., Smith equivalent representations of generalized quaternion groups, in Group actions on manifolds (ed. Schultz, R.), Contemporary Mathematics, Volume 36, pp. 317–322 (American Mathematical Society, Providence, RI, 1985).CrossRefGoogle Scholar
8.Cho, E. C. and Suh, D. Y., Induction in equivariant K-theory and s-Smith equivalence of representations, in Group actions on manifolds (ed. Schultz, R.), Contemporary Mathematics, Volume 36, pp. 311–315 (American Mathematical Society, Providence, RI, 1985).CrossRefGoogle Scholar
9.Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., Atlas of finite groups: maximal subgroups and ordinary characters for simple groups (Oxford University Press, 1985).Google Scholar
10.Dovermann, K. H. and Herzog, M., Gap conditions for representations of symmetric groups, J. Pure Appl. Alg. 119 (1987), 113–137.CrossRefGoogle Scholar
11.Dovermann, K. H. and Petrie, T., Smith equivalence of representations for odd order cyclic groups, Topology 24 (1985), 283–305.CrossRefGoogle Scholar
12.Dovermann, K. H. and Washington, L. D., Relations between cyclotomic units and Smith equivalence of representations, Topology 28 (1989), 81–89.CrossRefGoogle Scholar
13.Dovermann, K. H., Petrie, T. and Schultz, R., Transformation groups and fixed point data, in Group actions on manifolds (ed. Schultz, R.), Contemporary Mathematics, Volume 36, pp. 159–189 (American Mathematical Society, Providence, RI, 1985).CrossRefGoogle Scholar
14.Dovermann, K. H., and Suh, D. Y., Smith equivalence for finite abelian groups, Pac. J. Math. 152 (1992), 41–78.CrossRefGoogle Scholar
15.Fulton, W. and Harris, J., Representation theory: a first course, Graduate Texts in Mathematics, Volume 129 (Springer, 1991).Google Scholar
16. GAP Group, The, GAP: Groups, Algorithms, and Programming, Version 4.4 (2006; available at www.gap-system.org).Google Scholar
17.Gorenstein, D., Lyons, R. and Solomon, R., The classification of the finite simple groups, Volume 1, American Mathematical Society Mathematical Surveys and Monographs, Volume 40 (American Mathematical Society, Providence, RI, 1994).CrossRefGoogle Scholar
18.Illman, S., Representations at fixed points of actions of finite groups on spheres, in Current trends in algebraic topology, Conference Proceedings, Canadian Mathematical Society, Volume 2, Part 2, pp. 135–155 (American Mathematical Society, Providence, RI, 1982).Google Scholar
19.James, G. and Liebeck, M., Representations and characters of groups, 2nd edn (Cambridge University Press, 2001).CrossRefGoogle Scholar
20.Ju, X. M., The Smith set of the group S 5 × C 2 × … × C 2, Osaka J. Math. 47 (2010), 215–236.Google Scholar
21.Kawakubo, K., The theory of transformation groups (Oxford University Press, 1991).CrossRefGoogle Scholar
22.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.Google Scholar
23.Laitinen, E. and Morimoto, M., Finite groups with smooth one fixed point actions on spheres, Forum Math. 10 (1998), 479–520.CrossRefGoogle Scholar
24.Laitinen, E. and Pawałowski, K., Smith equivalence of representations for finite perfect groups, Proc. Am. Math. Soc. 127 (1999), 297–307.CrossRefGoogle Scholar
25.Laitinen, E., Morimoto, M. and Pawałowski, K., Deleting-inserting theorem for smooth actions of finite non-solvable groups on spheres, Comment. Math. Helv. 70 (1995), 10–38.CrossRefGoogle Scholar
26.Masuda, M., and Petrie, T., Lectures on transformation groups and Smith equivalence, in Group actions on manifolds (ed. Schultz, R.), Contemporary Mathematics, Volume 36, pp. 191–242 (American Mathematical Society, Providence, RI, 1985).CrossRefGoogle Scholar
27.Milnor, J. W., Whitehead torsion, Bull. Am. Math. Soc. 72 (1966), 358–426.CrossRefGoogle Scholar
28.Morimoto, M., Equivariant surgery theory: deleting-inserting theorem of fixed point manifolds on spheres and disks, K-Theory 15 (1998), 13–32.CrossRefGoogle Scholar
29.Morimoto, M., Smith equivalent Aut(A 6)-representations are isomorphic, Proc. Am. Math. Soc. 136 (2008), 3683–3688.CrossRefGoogle Scholar
30.Morimoto, M., Nontrivial (G)-matched -related pairs for finite gap Oliver groups, J. Math. Soc. Jpn 62(2) (2010), 623–647.Google Scholar
31.Morimoto, M. and Pawałowski, K., The equivariant bundle subtraction theorem and its applications, Fund. Math. 161 (1999), 279–303.CrossRefGoogle Scholar
32.Morimoto, M. and Pawałowski, K., Smooth actions of finite Oliver groups on spheres, Topology 42 (2003), 395–421.CrossRefGoogle Scholar
33.Morimoto, M. and Qi, Y., Study of Smith sets of gap Oliver groups, in Transformation groups from a new viewpoint, RIMS Kokyuroku, Volume 1670, 126–139 (Kyoto University, 2009).Google Scholar
34.Morimoto, M. and Qi, Y., The primary Smith sets of finite Oliver groups, in Group actions and homogeneous spaces, Proc. Bratislava Topology Symp., 7–11 September 2009, p. 61–73 (Comenius University, Bratislava, 2010).Google Scholar
35.Morimoto, M., Sumi, T. and Yanagihara, M., Finite groups possessing gap modules, in Geometry and topology: Aarhus (ed. Grove, K., Madsen, I. H. and Pedersen, E. K.), Contemporary Mathematics, Volume 258, pp. 329–342 (American Mathematical Society, Providence, RI, 2000).CrossRefGoogle Scholar
36.Oliver, B., Fixed point sets and tangent bundles of actions on disks and Euclidean spaces, Topology 35 (1996), 583–615.CrossRefGoogle Scholar
37.Oliver, R., Fixed point sets of group actions on finite acyclic complexes, Comment. Math. Helv. 50 (1975), 155–177.CrossRefGoogle Scholar
38.Pawałowski, K., Group actions with inequivalent representations at fixed points, Math. Z. 187 (1984), 29–47.CrossRefGoogle Scholar
39.Pawałowski, K., Smith equivalence of group modules and the Laitinen conjecture: a survey, in Geometry and topology: Aarhus (ed. Grove, K., Madsen, I. H., Pedersen, E. K.), Contemporary Mathematics, Volume 258, pp. 43–350 (American Mathematical Society, Providence, RI, 2000).Google Scholar
40.Pawałowski, K. and Solomon, R., Smith equivalence and finite Oliver groups with Laitinen number 0 or 1, Alg. Geom. Topology 2 (2002), 843–895.CrossRefGoogle Scholar
41.Pawałowski, K. and Sumi, T., Finite groups with Smith equivalent, nonisomorphic representations, in Proc. 33rd Symp. on Transformation Groups (ed. Kawakami, T.), pp. 68–76 (Wing, Wakayama, 2007).Google Scholar
42.Pawałowski, K. and Sumi, T., Smith equivalent group modules, in Proc. 34th Symp. on Transformation Groups (ed. Kawakami, T. and Yamasaki, M.), pp. 68–75 (Wing, Wakayama, 2007).Google Scholar
43.Pawałowski, K. and Sumi, T., The Laitinen Conjecture for finite solvable Oliver groups, Proc. Am. Math. Soc. 137 (2009), 2147–2156.CrossRefGoogle Scholar
44.Petrie, T., G sugery, I, A survey, in Algebraic and geometric topology, Lecture Notes in Mathematics, Volume 664, pp. 197–233 (Springer, 1978).CrossRefGoogle Scholar
45.Petrie, T., Pseudoequivalences of G-manifolds, Proc. Symp. Pure Math. 32 (1978), 169–210.CrossRefGoogle Scholar
46.Petrie, T., Three theorems in transformation groups, in Algebraic topology: Aarhus, Lecture Notes in Mathematics, Volume 763, pp. 549–572 (Springer, 1979).Google Scholar
47.Petrie, T., The equivariant J homomorphhism and Smith equivalence of representations, Current trends in algebraic topology, Conference Proceedings, Canadian Mathematical Society, Volume 2, Part 2, pp. 223–233 (American Mathematical Society, Providence, RI, 1982).Google Scholar
48.Petrie, T., Smith equivalence of representations, Math. Proc. Camb. Phil. Soc. 94 (1983), 61–99.CrossRefGoogle Scholar
49.Petrie, T. and Randall, J., Transformation groups on manifolds, Monographs and Textbooks in Pure and Applied Mathematics, Volume 82 (Dekker, New York, 1984).Google Scholar
50.Petrie, T. and Randall, J., Spherical isotropy representations, Publ. Math. IHES 62 (1985), 5–40.CrossRefGoogle Scholar
51.Sanchez, C. U., Actions of groups of odd order on compact orientable manifolds, Proc. Am. Math. Soc. 54 (1976), 445–448.CrossRefGoogle Scholar
52.Schultz, R., Problems submitted to the AMS Summer Research Conference on Group Actions, collected and edited by Schultz, R., in Group actions on manifolds (ed. Schultz, R.), Contemporary Mathematics, Volume 36, pp. 513–568 (American Mathematical Society, Providence, RI, 1985).CrossRefGoogle Scholar
53.Serre, J.-P., Linear representations of finite groups, Graduate Texts in Mathematics, Volume 42 (Springer, 1977).CrossRefGoogle Scholar
54.Smith, P. A., New results and old problems in finite transformation groups, Bull. Am. Math. Soc. 66 (1960), 401–415.CrossRefGoogle Scholar
55.Steinberg, R., The representations of GL(3, q), GL(4, q), PGL(3, q) and PGL(4, q), Can. J. Math. 3 (1951), 225–235.CrossRefGoogle Scholar
56.Suh, D. Y., s-Smith equivalent representations of finite abelian groups, in Group actions on manifolds (ed. Schultz, R.), Contemporary Mathematics, Volume 36, pp. 323–329 (American Mathematical Society, Providence, RI, 1985).CrossRefGoogle Scholar
57.Sumi, T., Gap modules for direct product groups, J. Math. Soc. Jpn 53 (2001), 975–990.CrossRefGoogle Scholar
58.Sumi, T., Gap modules for semidirect product groups, Kyushu J. Math. 58 (2004), 33–58.CrossRefGoogle Scholar
59.Sumi, T., Finite groups possessing Smith equivalent, nonisomorphic representations, in The theory of transformation groups and its representations, RIMS Kokyuroku, Volume 1569, pp. 170–179 (Kyoto University, 2007).Google Scholar
60.Sumi, T., Smith problem for a finite Oliver group with non-trivial center, in Geometry of transformation groups and related topics, RIMS Kokyuroku, Volume 1612, pp. 196–204 (Kyoto University, 2008).Google Scholar
61.Sumi, T., Smith set for a nongap Oliver group, in Transformation groups from a new viewpoint, RIMS Kokyuroku, Volume 1670, pp. 25–33 (Kyoto University, 2009).Google Scholar
62.Sumi, T., Representation spaces fulfilling the gap hypothesis, in Group Actions and Homogeneous Spaces, Proc. Bratislava Topology Symp., 7–11 September 2009, pp. 99–116 (Comenius University, Bratislava, 2010).Google Scholar
63.Sumi, T., The gap hypothesis for finite groups which have an abelian quotient group not of order a power of 2, J. Math. Soc. Jpn 64 (2012), 91–106.CrossRefGoogle Scholar
64.Dieck, T. tom, Transformation groups, de Gruyter Studies in Mathematics, Volume 8 (Walter de Gruyter, 1987).CrossRefGoogle Scholar