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Tytuł artykułu

The Equivariant Bundle Subtraction Theorem and its applications

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In the theory of transformation groups, it is important to know what kind of isotropy subgroups of G do occur at points of the space upon which the given group G acts. In this article, for a finite group G, we prove the Equivariant Bundle Subtraction Theorem (Theorem 2.2) which allows us to construct smooth G-manifolds with prescribed isotropy subgroups around the G-fixed point sets. In Theorem 0.1, we restate Oliver's result about manifolds M and G-vector bundles over M that occur, respectively, as the G-fixed point sets and their equivariant normal bundles for smooth G-actions on disks. In Theorems 0.2 and 0.3, we prove the corresponding results for smooth G-actions on disks with prescribed isotropy subgroups around M. In Theorems 0.4 and 0.5, for large classes of finite groups G, we explicitly describe manifolds M that occur as the G-fixed point sets for such actions on disks. These actions are expected to be useful for answering the question of which manifolds occur as the G-fixed points sets for smooth G-actions on spheres.
Twórcy
  • Department of Mathematical and Environmental Sciences, Faculty of Environmental Science and Technology, Okayama University, Tsushimanaka 2-1-1 Okayama, 700 Japan, morimoto@math.ems.okayama-u.ac.jp
  • Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Jana Matejki 48/49, 60-769 Poznań, Poland, kpa@math.amu.edu.pl
Bibliografia
  • [BM] A. Bak and M. Morimoto, Equivariant surgery with middle dimensional singular sets. I, Forum Math. 8 (1996), 267-302.
  • [Br] G. E. Bredon, Introduction to Compact Transformation Groups, Pure Appl. Math. 46, Academic Press, New York, 1972.
  • [tD] T. tom Dieck, Transformation Groups, de Gruyter Stud. in Math. 8, Walter de Gruyter, 1987.
  • [EL] A. L. Edmonds and R. Lee, Fixed point sets of group actions on Euclidean space, Topology 14 (1975), 339-345.
  • [H] D. Husemoller, Fibre Bundles, 3rd ed., Grad. Texts in Math. 20, Springer, New York, 1994.
  • [K] K. Kawakubo, The Theory of Transformation Groups, Oxford Univ. Press, Oxford, 1991.
  • [LM] E. Laitinen and M. Morimoto, Finite groups with smooth one fixed point actions on spheres, Forum Math. 10 (1998), 479-520.
  • [LMP] 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.
  • [M1] M. Morimoto, Positioning map, equivariant surgery obstruction, and applications, Kyoto University RIMS Kokyuroku 793 (1992), 75-93.
  • [M2] M. Morimoto, Equivariant surgery theory: Construction of equivariant normal maps, Publ. Res. Inst. Math. Sci. 31 (1995), 145-167.
  • [M3] M. Morimoto, Equivariant surgery theory: Deleting-Inserting Theorem of fixed point manifolds on spheres and disks, K-Theory 15 (1998), 13-32.
  • [N] T. Nakayama, On modules of trivial cohomology over a finite group, II (Finitely generated modules), Nagoya Math. J. Ser. A 12 (1957), 171-176.
  • [O1] R. Oliver, Fixed point sets of group actions on finite acyclic complexes, Comment. Math. Helv. 50 (1975), 155-177.
  • [O2] R. Oliver, Smooth compact Lie group actions on disks, Math. Z. 149 (1976), 79-96.
  • [O3] R. Oliver, G-actions on disks and permutation representations II, ibid. 157 (1977), 237-263.
  • [O4] R. Oliver, Fixed point sets and tangent bundles of actions on disks and euclidean spaces, Topology 35 (1996), 583-615.
  • [P1] K. Pawałowski, Fixed point sets of smooth group actions on disks and Euclidean spaces. A survey, in: Geometric and Algebraic Topology, Banach Center Publ. 18, PWN, Warszawa, 1986, 165-180.
  • [P2] K. Pawałowski, Fixed point sets of smooth group actions on disks and Euclidean spaces, Topology 28 (1989), 273-289; Corrections: ibid. 35 (1996), 749-750.
  • [P3] K. Pawałowski, Chern and Pontryagin numbers in perfect symmetries of spheres, K-Theory 13 (1998), 41-55.
  • [R] D. S. Rim, Modules over finite groups, Ann. of Math. 69 (1959), 700-712.
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Bibliografia
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bwmeta1.element.bwnjournal-article-fmv161i3p279bwm
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