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Fixed-point sets of smooth actions on spheres

Published online by Cambridge University Press:  30 November 2007

Masaharu Morimoto
Masaharu Morimoto
Affiliation:
morimoto@ems.okayama-u.ac.jpDepartment of Environmental and Mathematical Sciences, Faculty of Environmental Science and Technology, Okayama University, Tsushimanaka 3-1-1, Okayama, 700-8530Japan
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Abstract

Given a group, it is a basic problem to determine which manifolds can occur as a fixed-point set of a smooth action of this group on a sphere. The current article answers this problem for a family of finite groups including perfect groups and nilpotent Oliver groups. We obtain the answer as an application of a new deleting and inserting theorem which is formulated to delete (or insert) fixed-point sets from (or to) disks with smooth actions of finite groups. One of the keys to the proof is an equivariant interpretation of the surgery theory of S. E. Cappell and J. L. Shaneson, for obtaining homology equivalences.

Keywords

Equivariant surgerysurgery obstructionstrong gap conditionfixed pointsphere
Type
Research Article
Information
Journal of K-Theory , Volume 1 , Issue 1 , February 2008 , pp. 95 - 128
Copyright
Copyright © ISOPP 2008

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References

1.Bak, A., K-theory of Forms, Annals of Math. Studies 98, Princeton Univ. Press, Princeton, 1981Google Scholar
2.Bak, A. and Morimoto, M., Equivariant surgery with middle dimensional singular sets. I, Forum Math. 8 (1996), 267–302CrossRefGoogle Scholar
3.Cappell, S. E. and Shaneson, J. L., The codimension two placement problem and homology equivalent manifolds, Ann. of Math. 99 (1974), 277–348CrossRefGoogle Scholar
4.tom Dieck, T., Transformation Groups, de Gruyter Studies in Math. 8, Walter de Gruyter, Berlin, 1979Google Scholar
5.Dress, A., Induction and structure theorems for Grothendieck and Witt rings of orthogonal representations of finite groups, Bull. Amer. Math. Soc. 79 (1973), 741–745CrossRefGoogle Scholar
6.Dress, A., Induction and structure theorems for orthogonal representations of finite groups, Ann. of Math. 102 (1975), 291–325CrossRefGoogle Scholar
7.Edmonds, A. and Lee, R., Fixed point sets of smooth group actions on disks and Euclidean spaces, Topology 14 (1975), 339–345CrossRefGoogle Scholar
8.Jones, L., The converse to the fixed point theorem of P. A. Smith: I, Annals of Math. 94 (1971), 52–68CrossRefGoogle Scholar
9.Ju, X.M., Matsuzaki, K. and Morimoto, M., Mackey and Frobenius structures on odd dimensional surgery obstruction groups, K-Theory 29 (2003), 285–312CrossRefGoogle Scholar
10.Laitinen, E. and Morimoto, M., Finite groups with smooth one fixed point actions on spheres, Forum Math. 10 (1998), 479–520CrossRefGoogle Scholar
11.Laitinen, E., Morimoto, M. and Pawałowski, K., Deleting-inserting theorem for smooth actions of finite solvable groups on spheres, Comment. Math. Helv. 70 (1995), 10–38CrossRefGoogle Scholar
12.Morimoto, M., Bak groups and equivariant surgery, K-Theory 2 (1989), 465–483CrossRefGoogle Scholar
13.Morimoto, M., Equivariant surgery theory: Construction of equivariant normal maps, Publ. Res. Inst. Math. Sci. Kyoto Univ. 31 (1995), 145–167CrossRefGoogle Scholar
14.Morimoto, M., Equivariant surgery theory: Deleting-inserting theorems of fixed point manifolds on spheres and disks, K-Theory 15 (1998), 13–32CrossRefGoogle Scholar
15.Morimoto, M., The Burnside ring revisited, in: Current Trends in Transformation Groups, eds. Bak, A., Morimoto, M. and Ushitaki, F., K-Monographs in Mathematics 7, pp.129–145, Kluwer Academic Publishers, Dordrecht-Boston-London, 2002Google Scholar
16.Morimoto, M., Induction theorems of surgery obstruction groups, Trans. Amer. Math. Soc. 355 (2003), 2341–2384CrossRefGoogle Scholar
17.Morimoto, M., Equivariant surgery theory for homology equivalences under the gap condition, Publ. Res. Inst. Math. Sci. Kyoto Univ. 42 (2006), 481–506CrossRefGoogle Scholar
18.Morimoto, M., Smith equivalent Aut(A 6)-representations are isomorphic, to appear in Proc. Amer. Math. Soc.Google Scholar
19.Morimto, M. and Iizuka, K., Extendibility of G-maps to pseudo-equivalences to finite G-CW-complexes whose fundamental groups are finite, Osaka J. Math. 21 (1984), 56–69Google Scholar
20.Morimoto, M. and Pawałowski, K., Equivariant wedge sum construction of finite contractible G-CW complexes with G-vector bundles, Osaka J. Math. 36 (1999), 767–781Google Scholar
21.Morimoto, M. and Pawałowski, K., The equivariant bundle subtraction theorem and its applications, Fundam. Math. 161 (1999), 279–303CrossRefGoogle Scholar
22.Morimoto, M. and Pawałowski, K., Smooth actions of finite Oliver groups on spheres, Topology 42 (2003), 395–421CrossRefGoogle Scholar
23.Morimoto, M., Sumi, T. and Yanagihara, M., Finite groups possessing gap modules, in: Geometry and Topology, Aarhus 1998, eds. Grove, K., Madsen, I. H. and Pedersen, E. K., pp.329–342, Contemp. Math. 258, Amer. Math. Soc., Providence, 2000Google Scholar
24.Oliver, B., Fixed point sets and tangent bundles of actions on disks and euclidean spaces, Topology 35 (1996), 583–615CrossRefGoogle Scholar
25.Oliver, R., Fixed point sets of group actions on finite acyclic complexes, Comment. Math. Helv. 50 (1975), 155–177CrossRefGoogle Scholar
26.Oliver, R. and Petrie, T., G-CW-surgery and K0(ℤ[G]), Math. Z. 179 (1982), 11–42CrossRefGoogle Scholar
27.Pawałowski, K., Manifolds as fixed point sets of smooth compact Lie group actions, in: Current Trends in Transformation Groups, eds. Bak, A., Morimoto, M. and Ushitaki, F., K-Monographs in Mathematics 7, pp.79–104, Kluwer Academic Publishers, Dordrecht-Boston-London, 2002Google Scholar
28.Ranicki, A., Algebraic L-theory. I. Foundations Proc. London Math. Soc. (3) 27 (1973), 101–125CrossRefGoogle Scholar
29.Smith, P.A., Fixed-points of periodic transformations, Amer. Math. Soc. Coll. Pub. 27 (1942), 350–373Google Scholar
30.Sumi, T., Gap modules for direct product groups, J. Math. Soc. Japan 53 (2001), 975–990CrossRefGoogle Scholar
31.Swan, R. G., Induced representations and projective modules, Ann. of Math. 71 (1960), 552–578CrossRefGoogle Scholar
32.Thomas, C. B., Frobenius reciprocity of Hermitian forms, J. Algebra 18 (1971), 237–244CrossRefGoogle Scholar
33.Wall, C. T. C., Surgery on Compact Manifolds, Academic Press, London–New York, 1970Google Scholar