## Osaka Journal of Mathematics

### One-fixed-point actions on spheres and Smith sets

Masaharu Morimoto

#### Abstract

Let $G$ be a finite group. The Smith equivalence for real $G$-modules of finite dimension gives a subset of real representation ring, called the primary Smith set. Since the primary Smith set is not additively closed in general, it is an interesting problem to find a subset which is additively closed in the real representation ring and occupies a large portion of the primary Smith set. In this paper we introduce an additively closed subset of the primary Smith set by means of smooth one-fixed-point $G$-actions on spheres, and we give evidences that the subset occupies a large portion of the primary Smith set if $G$ is an Oliver group.

#### Article information

Source
Osaka J. Math., Volume 53, Number 4 (2016), 1003-1013.

Dates
First available in Project Euclid: 4 October 2016

https://projecteuclid.org/euclid.ojm/1475601828

Mathematical Reviews number (MathSciNet)
MR3554853

Zentralblatt MATH identifier
1361.57039

#### Citation

Morimoto, Masaharu. One-fixed-point actions on spheres and Smith sets. Osaka J. Math. 53 (2016), no. 4, 1003--1013. https://projecteuclid.org/euclid.ojm/1475601828

#### References

• A. Bak and M. Morimoto: The dimension of spheres with smooth one fixed point actions, Forum Math. 17 (2005), 199–216.
• G.E. Bredon: Representations at fixed points of smooth actions of compact groups, Ann. of Math. (2) 89 (1969), 515–532.
• S.E. Cappell and J.L. Shaneson: Fixed points of periodic differentiable maps, Invent. Math. 68 (1982), 1–19.
• S.E. Cappell and J.L. Shaneson: Representations at fixed points; in Group Actions on Manifolds (Boulder, Colo., 1983), Contemp. Math. 36, Amer. Math. Soc., Providence, RI, 151–158, 1985.
• \begingroup A. Koto, M. Morimoto and Y. Qi: The Smith sets of finite groups with normal Sylow 2-subgroups and small nilquotients, J. Math. Kyoto Univ. 48 (2008), 219–227. \endgroup
• E. Laitinen and M. Morimoto: Finite groups with smooth one fixed point actions on spheres, Forum Math. 10 (1998), 479–520.
• E. Laitinen, M. Morimoto and K. Pawałowski: Deleting-inserting theorem for smooth actions of finite nonsolvable groups on spheres, Comment. Math. Helv. 70 (1995), 10–38.
• E. Laitinen and K. Pawałowski: Smith equivalence of representations for finite perfect groups, Proc. Amer. Math. Soc. 127 (1999), 297–307.
• M. Morimoto: Smith equivalent $\mathrm{Aut}(A_{6})$-representations are isomorphic, Proc. Amer. Math. Soc. 136 (2008), 3683–3688.
• M. Morimoto: Nontrivial $\mathcal{P}(G)$-matched $\mathfrak{S}$-related pairs for finite gap Oliver groups, J. Math. Soc. Japan 62 (2010), 623–647.
• M. Morimoto: Deleting and inserting fixed point manifolds under the weak gap condition, Publ. Res. Inst. Math. Sci. 48 (2012), 623–651.
• \begingroup M. Morimoto: Tangential representations of one-fixed-point actions on spheres and Smith equivalence, J. Math. Soc. Japan 67 (2015), 195–205. \endgroup
• M. Morimoto: A necessary condition for the Smith equivalence of $G$-modules and its sufficiency, to appear in Math. Slov.
• M. Morimoto and K. Pawałowski: Smooth actions of finite Oliver groups on spheres, Topology 42 (2003), 395–421.
• M. Morimoto and Y. Qi: The primary Smith sets of finite Oliver groups; in Group Actions and Homogeneous Spaces, Fak. Mat. Fyziky Inform. Univ. Komenského, Bratislava, 61–73, 2010.
• M. Morimoto, T. Sumi and M. Yanagihara: Finite groups possessing gap modules; in Geometry and Topology: Aarhus (1998), Contemp. Math. 258, Amer. Math. Soc., Providence, RI, 329–342, 2000.
• B. Oliver: Fixed point sets and tangent bundles of actions on disks and Euclidean spaces, Topology 35 (1996), 583–615.
• R. Oliver: Fixed-point sets of group actions on finite acyclic complexes, Comment. Math. Helv. 50 (1975), 155–177.
• K. Pawałowski and R. Solomon: Smith equivalence and finite Oliver groups with Laitinen number $0$ or $1$, Algebr. Geom. Topol. 2 (2002), 843–895.
• K. Pawałowski and T. Sumi: The Laitinen conjecture for finite solvable Oliver groups, Proc. Amer. Math. Soc. 137 (2009), 2147–2156.
• K. Pawałowski and T. Sumi: The Laitinen conjecture for finite non-solvable groups, Proc. Edinb. Math. Soc. (2) 56 (2013), 303–336.
• T. Petrie and J.D. Randall: Transformation Groups on Manifolds, Monographs and Textbooks in Pure and Applied Mathematics 82, Dekker, New York, 1984.
• Y. Qi: The tangent bundles over equivariant real projective spaces, Math. J. Okayama Univ. 54 (2012), 87–96.
• Y. Qi: The canonical line bundles over equivariant real projective spaces, Math. J. Okayama Univ. 57 (2015), 111–122.
• C.U. Sanchez: Actions of groups of odd order on compact, orientable manifolds, Proc. Amer. Math. Soc. 54 (1976), 445–448.
• J.-P. Serre: Linear Representations of Finite Groups, Graduate Texts in Mathematics 42, Springer, New York, 1977.
• P.A. Smith: New results and old problems in finite transformation groups, Bull. Amer. Math. Soc. 66 (1960), 401–415.
• T. Sumi: Gap modules for direct product groups, J. Math. Soc. Japan 53 (2001), 975–990.
• T. Sumi: The gap hypothesis for finite groups which have an abelian quotient group not of order a power of 2, J. Math. Soc. Japan 64 (2012), 91–106.
• T. Sumi: Centralizers of gap groups, Fund. Math. 226 (2014), 101–121.