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Smith equivalence and finite Oliver groups with Laitinen number 0 or 1. (English) Zbl 1022.57019
Based on a question of P. A. Smith, two representations $$V$$ and $$W$$ of a finite (or compact Lie) group $$G$$ are called Smith equivalent if there exists a smooth action of $$G$$ on a (homotopy) sphere $$\Sigma$$ with exactly two fixed points, and the tangent representations at these fixed points are $$V$$ and $$W$$, respectively. It is a delicate matter to show that there are non-isomorphic Smith equivalent representations, and to construct $$\Sigma$$ as above. Some difficulties can be contained if one considers only Oliver groups, groups which can act on a disk without a $$G$$-fixed point. An additional invariant $$\alpha_G$$ of the group $$G$$, called Laitinen number, provides additional insight into the possibility of constructing non-isomorphic Smith equivalent representations. The body of the paper is a careful group theoretic classification of Oliver groups with Laitinen number at most $$1$$. Several authors have provided non-isomorphic Smith equivalent representations for a variety of groups. The Laitinen number $$\alpha_G$$ being $$0$$ or $$1$$ is a good indication that Smith equivalent representations must be isomorphic. For some classes of groups this is a theorem. If $$\alpha_G \geq 2$$, then one expects that there are non-isomorphic Smith equivalent representations of $$G$$. Again, this is a theorem with one possible exception and valid for a certain class of groups only.

##### MSC:
 57S17 Finite transformation groups 55M35 Finite groups of transformations in algebraic topology (including Smith theory) 57S25 Groups acting on specific manifolds 20D05 Finite simple groups and their classification
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