### 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.

Original language | English |
---|---|

Pages (from-to) | 95-128 |

Number of pages | 34 |

Journal | Journal of K-Theory |

Volume | 1 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2008 |

### Fingerprint

### Keywords

- Equivariant surgery
- fixed point
- sphere
- strong gap condition
- surgery obstruction

### ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology

### Cite this

**Fixed-point sets of smooth actions on spheres.** / Morimoto, Masaharu.

Research output: Contribution to journal › Article

*Journal of K-Theory*, vol. 1, no. 1, pp. 95-128. https://doi.org/10.1017/is007011012jkt009

}

TY - JOUR

T1 - Fixed-point sets of smooth actions on spheres

AU - Morimoto, Masaharu

PY - 2008

Y1 - 2008

N2 - 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.

AB - 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.

KW - Equivariant surgery

KW - fixed point

KW - sphere

KW - strong gap condition

KW - surgery obstruction

UR - http://www.scopus.com/inward/record.url?scp=85012559698&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85012559698&partnerID=8YFLogxK

U2 - 10.1017/is007011012jkt009

DO - 10.1017/is007011012jkt009

M3 - Article

AN - SCOPUS:85012559698

VL - 1

SP - 95

EP - 128

JO - Journal of K-Theory

JF - Journal of K-Theory

SN - 1865-2433

IS - 1

ER -