Smith equivalence of representations

for finite perfect groups

Authors:
Erkki Laitinen and Krzysztof Pawalowski

Journal:
Proc. Amer. Math. Soc. **127** (1999), 297-307

MSC (1991):
Primary 57S17, 57S25

DOI:
https://doi.org/10.1090/S0002-9939-99-04544-X

MathSciNet review:
1468195

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Using smooth one-fixed-point actions on spheres and a result due to Bob Oliver on the tangent representations at fixed points for smooth group actions on disks, we obtain a similar result for perfect group actions on spheres. For a finite group , we compute a certain subgroup of the representation ring . This allows us to prove that a finite perfect group has a smooth -proper action on a sphere with isolated fixed points at which the tangent representations of are mutually nonisomorphic if and only if contains two or more real conjugacy classes of elements not of prime power order. Moreover, by reducing group theoretical computations to number theory, for an integer and primes , we prove similar results for the group , , or . In particular, has Smith equivalent representations that are not isomorphic if and only if , , .

**[AB]**Atiyah, M.F. and Bott,R.,*A Lefschetz fixed point formula for elliptic complexes: II. Applications*, Ann. of Math.**88**(1968), 451-491. MR**38:731****[Br]**Bredon, G.,*Representations at fixed points of smooth actions of compact groups*, Ann. of Math.**89**(1969), 515-532. MR**39:7628****[CS1]**Cappell, S.E. and Shaneson, J.L.,*Fixed points of periodic maps*, Proc. Nat. Acad. Sci. USA**77**(1980), 5052-5054. MR**81m:57029****[CS2]**Cappell, S.E. and Shaneson, J.L.,*Fixed points of periodic differentiable maps*, Invent. Math.**68**(1982), 1-19. MR**84a:57038****[CS3]**Cappell, S.E. and Shaneson, J.L.,*Representations at fixed points*, in Group Actions on Manifolds (ed. R.Schultz), Contemp. Math.**36**(1985), 151-158. MR**86h:57038****[DP]**Dovermann, K.H. and Petrie, T.,*Smith equivalence of representations for odd order cyclic groups*, Topology**24**(1985), 283-305. MR**87d:57030****[DS]**Dovermann, K.H. and Suh, D.Y.,*Smith equivalence for finite abelian groups*, Pacific J. Math.**152**(1992), 41-78. MR**93d:57069****[DW]**Dovermann, K.H. and Washington, L.D.,*Relations between cyclotomic units and Smith equivalence of representations*, Topology**28**(1989), 81-89. MR**90e:57071****[DPS]**Dovermann, K.H., Petrie, T. and Schultz, R.,*Transformation groups and fixed point data*, in Group Actions on Manifolds (ed. R. Schultz), Contemp. Math.**36**(1985), 159-189. MR**86e:57039****[HW]**Hardy, G. H. and Wright, E. M.,*An Introduction to the Theory of Numbers, 3rd Ed.*, Oxford University Press, London, 1954. MR**16:673c****[H]**Huppert, B.,*Endliche Gruppen I*, Grundlehren 134, Springer, Berlin, 1967. MR**37:302****[LM]**Laitinen, E. and Morimoto, M.,*Finite groups with smooth one fixed point actions on spheres*, Forum Math.**10**(1998), 479-520.**[LMP]**Laitinen, E., Morimoto, M. and Pawa{\l}owski, K.,*Deleting-Inserting Theorem for smooth actions of finite nonsolvable groups on spheres*, Comment. Math. Helv.**70**(1995), 10-38. MR**96b:57043****[MP]**Masuda, M. and Petrie, T.,*Lectures on transformation groups and Smith equivalence*, in Group Actions and Manifolds (ed. R. Schultz), Contemp. Math.**36**(1985), 191-242. MR**87a:57047****[Mi]**Milnor, J.W.,*Whitehead torsion*, Bull. Amer. Math. Soc.**72**(1966), 358-426. MR**33:4922****[O]**Oliver, B.,*Fixed point sets and tangent bundles of actions on disks and Euclidean spaces*, Topology**35**(1996), 583-615. MR**97g:57059****[Pe1]**Petrie, T.,*-surgery I - A survey*, in Algebraic and Geometric Topology, Proceedings (Santa Barbara 1977), Lecture Notes in Math.**664**(1978), 197-233. MR**80g:57049****[Pe2]**Petrie, T.,*Pseudoequivalences of -manifolds*, in Algebraic and Geometric Topology, Proc. Symp. in Pure Math.**32**(1978), 169-210. MR**80e:57039****[Pe3]**Petrie, T.,*Smith equivalence of representations*, Math. Proc. Cambridge Philos. Soc.**94**(1983), 61-99. MR**85i:57011****[PR]**Petrie, T. and Randall, J.,*Spherical isotropy representations*, Publ. Math. IHES**62**(1985), 5-40. MR**88i:57017****[Sa]**Sanchez, C.U.,*Actions of groups of odd order on compact orientable manifolds*, Proc. Amer. Math. Soc.**54**(1976), 445-448. MR**53:11641****[Sch]**Schultz, R., Problems submitted to the A.M.S. Summer Research Conference on Group Actions, Collected and edited by R. Schultz, in*Group Actions on Manifolds*, Contemp. Math., vol. 36, 1985, pp. 513-568. MR**86i:57042****[Se1]**Serre, J.-P.,*Cours d'Arithmétique*, Presses Universitaires de France, Paris, 1970. MR**41:138****[Se2]**Serre, J.-P.,*Linear Representations of Finite Groups*, GTM 42, Springer, New York, 1977. MR**56:8675****[Sm]**Smith, P.A.,*New results and old problems in finite transformation groups*, Bull. Amer. Math. Soc.**66**(1960), 401-415. MR**23:A2880**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
57S17,
57S25

Retrieve articles in all journals with MSC (1991): 57S17, 57S25

Additional Information

**Erkki Laitinen**

Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University of Poznań, ul. Jana Matejki 48/49, PL–60–769 Poznań, Poland

Email:
kpa@math.amu.edu.pl

**Krzysztof Pawalowski**

Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University of Poznań, ul. Jana Matejki 48/49, PL–60–769 Poznań, Poland

DOI:
https://doi.org/10.1090/S0002-9939-99-04544-X

Keywords:
Finite perfect group,
action on sphere,
Smith equivalence of representations

Received by editor(s):
August 30, 1996

Received by editor(s) in revised form:
May 10, 1997

Communicated by:
Thomas Goodwillie

Article copyright:
© Copyright 1999
American Mathematical Society