On one fixed point actions on spheres. (English) Zbl 0724.57026
The author investigates smooth actions of finite groups on spheres having exactly one fixed point. Proofs of the following results are outlined.
Theorem A. There is an action of the alternating group \(A_ 5\) on \(S^ 6\) having one fixed point.
Theorem B. Let \(\Xi\) be a 4-dimensional integral homology sphere on which a finite group G acts. Let \(\Xi^ G_ h\) be the set of h-dimensional connected components of the fixed point set \(\Xi^ G\). Then \(| \Xi^ G_ 0| \leq 2.\)
Using a (yet unpublished) result of M. Furuka, Theorem B can be sharpened as follows:
Theorem C. If \(\Xi\) is a 4-dimensional homotopy sphere and \(\Xi^ G_ 0\neq \emptyset\) then \(\Xi^ G\) consists of exactly 2 points.
Theorem A. There is an action of the alternating group \(A_ 5\) on \(S^ 6\) having one fixed point.
Theorem B. Let \(\Xi\) be a 4-dimensional integral homology sphere on which a finite group G acts. Let \(\Xi^ G_ h\) be the set of h-dimensional connected components of the fixed point set \(\Xi^ G\). Then \(| \Xi^ G_ 0| \leq 2.\)
Using a (yet unpublished) result of M. Furuka, Theorem B can be sharpened as follows:
Theorem C. If \(\Xi\) is a 4-dimensional homotopy sphere and \(\Xi^ G_ 0\neq \emptyset\) then \(\Xi^ G\) consists of exactly 2 points.
Reviewer: H.M.Unsöld (Berlin)
MSC:
| 57S25 | Groups acting on specific manifolds |
| 55M35 | Finite groups of transformations in algebraic topology (including Smith theory) |
Keywords:
smooth actions; finite groups; spheres; fixed point; alternating group; homology sphere; homotopy sphereReferences:
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