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The Laitinen conjecture for finite solvable Oliver groups. (English) Zbl 1173.57015

The paper under review studies Smith’s problem, saying when the tangential representations at two isolated fixed points of a smooth action of a finite group on a sphere are linearly equivalent. The problem has been answered in many cases. The Laitinen conjecture is stated as follows: For any finite Oliver group \(G\) with \(r_G \geq 2\), there exist two Smith equivalent real \(G\)-modules which are not isomorphic, and the action of \(G\) on the homotopy sphere in question satisfies the Laitinen Condition: the fixed point set is connected for every \(g\in G\) of order \(2^k\) for \(k\geq 3\).
In [Proc. Am. Math. Soc. 136, No. 10, 3683–3688 (2008; Zbl 1151.55003)], M. Morimoto first gave a counterexample to the Laitinen conjecture for actions of the non-solvable group \(\text{Aut}(A_{6})\). The authors of the present paper obtain counterexamples for actions of five different solvable groups.
Reviewer: Zhi Lü (Shanghai)

MSC:

57S17 Finite transformation groups
57S25 Groups acting on specific manifolds

Citations:

Zbl 1151.55003

Software:

GAP
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References:

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