# The Equivariant Bundle Subtraction Theorem and its applications

Fundamenta Mathematicae (1999)

• Volume: 161, Issue: 3, page 279-303
• ISSN: 0016-2736

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## Abstract

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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.

## How to cite

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Morimoto, Masaharu, and Pawałowski, Krzysztof. "The Equivariant Bundle Subtraction Theorem and its applications." Fundamenta Mathematicae 161.3 (1999): 279-303. <http://eudml.org/doc/212407>.

@article{Morimoto1999,
abstract = {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.},
author = {Morimoto, Masaharu, Pawałowski, Krzysztof},
journal = {Fundamenta Mathematicae},
keywords = {equivariant bundle subtraction; smooth action on disk; fixed point set; equivariant normal bundle; the family of large subgroups of a finite group; large subgroups of a finite group},
language = {eng},
number = {3},
pages = {279-303},
title = {The Equivariant Bundle Subtraction Theorem and its applications},
url = {http://eudml.org/doc/212407},
volume = {161},
year = {1999},
}

TY - JOUR
AU - Morimoto, Masaharu
AU - Pawałowski, Krzysztof
TI - The Equivariant Bundle Subtraction Theorem and its applications
JO - Fundamenta Mathematicae
PY - 1999
VL - 161
IS - 3
SP - 279
EP - 303
AB - 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.
LA - eng
KW - equivariant bundle subtraction; smooth action on disk; fixed point set; equivariant normal bundle; the family of large subgroups of a finite group; large subgroups of a finite group
UR - http://eudml.org/doc/212407
ER -

## References

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1. [BM] A. Bak and M. Morimoto, Equivariant surgery with middle dimensional singular sets. I, Forum Math. 8 (1996), 267-302. Zbl0860.57028
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9. [M1] M. Morimoto, Positioning map, equivariant surgery obstruction, and applications, Kyoto University RIMS Kokyuroku 793 (1992), 75-93.
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12. [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. Zbl0207.33602
13. [O1] R. Oliver, Fixed point sets of group actions on finite acyclic complexes, Comment. Math. Helv. 50 (1975), 155-177. Zbl0304.57020
14. [O2] R. Oliver, Smooth compact Lie group actions on disks, Math. Z. 149 (1976), 79-96. Zbl0334.57023
15. [O3] R. Oliver, G-actions on disks and permutation representations II, ibid. 157 (1977), 237-263.
16. [O4] R. Oliver, Fixed point sets and tangent bundles of actions on disks and euclidean spaces, Topology 35 (1996), 583-615. Zbl0861.57047
17. [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.
18. [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. Zbl0691.57017
19. [P3] K. Pawałowski, Chern and Pontryagin numbers in perfect symmetries of spheres, K-Theory 13 (1998), 41-55. Zbl0897.57023
20. [R] D. S. Rim, Modules over finite groups, Ann. of Math. 69 (1959), 700-712. Zbl0092.26104

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