Abstract
Since 1946 it has been an open question which compact Lie groups can act smoothly on some sphere with exactly one fixed point. In this paper we solve the problem completely for finite groups: these groups are exactly those which can act smoothly on some disk without fixed points, a class determined by R. Oliver. Our main tools are the Burnside ring and the Grothendieck-Witt ring (classical to some extent) and a form of equivariant surgery theory allowing middle-dimensional singular sets developed recently.
| Original language | English |
|---|---|
| Pages (from-to) | 479-520 |
| Number of pages | 42 |
| Journal | Forum Mathematicum |
| Volume | 10 |
| Issue number | 4 |
| Publication status | Published - 1998 |
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ASJC Scopus subject areas
- Mathematics(all)
- Applied Mathematics
Cite this
Finite groups with smooth one fixed point actions on spheres. / Laitinen, Erkki; Morimoto, Masaharu.
In: Forum Mathematicum, Vol. 10, No. 4, 1998, p. 479-520.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Finite groups with smooth one fixed point actions on spheres
AU - Laitinen, Erkki
AU - Morimoto, Masaharu
PY - 1998
Y1 - 1998
N2 - Since 1946 it has been an open question which compact Lie groups can act smoothly on some sphere with exactly one fixed point. In this paper we solve the problem completely for finite groups: these groups are exactly those which can act smoothly on some disk without fixed points, a class determined by R. Oliver. Our main tools are the Burnside ring and the Grothendieck-Witt ring (classical to some extent) and a form of equivariant surgery theory allowing middle-dimensional singular sets developed recently.
AB - Since 1946 it has been an open question which compact Lie groups can act smoothly on some sphere with exactly one fixed point. In this paper we solve the problem completely for finite groups: these groups are exactly those which can act smoothly on some disk without fixed points, a class determined by R. Oliver. Our main tools are the Burnside ring and the Grothendieck-Witt ring (classical to some extent) and a form of equivariant surgery theory allowing middle-dimensional singular sets developed recently.
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UR - http://www.scopus.com/inward/citedby.url?scp=0032423724&partnerID=8YFLogxK
M3 - Article
AN - SCOPUS:0032423724
VL - 10
SP - 479
EP - 520
JO - Forum Mathematicum
JF - Forum Mathematicum
SN - 0933-7741
IS - 4
ER -