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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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