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Critical points of invariant functions on closed orientable surfaces. (English) Zbl 1321.55003

In this paper, the authors study orientation-preserving actions of finite groups on closed orientable surfaces. The first part of their work consists of a description of the gradient field of an equivariant \(C^1\)-function and of an elementary, differential proof of the Riemann-Hurwitz formula. In a second part, they consider the equivariant Lusternik-Schnirelmann category and prove that it equals, with a few exceptions, the number of singular orbits of the action. They also show that the upper bound given by H. Colman [Contemp. Math. 316, 35–40 (2002; Zbl 1035.55003)] is sharp in the case of preserving orientation actions on surfaces.

MSC:

55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
55M35 Finite groups of transformations in algebraic topology (including Smith theory)
30F10 Compact Riemann surfaces and uniformization

Citations:

Zbl 1035.55003
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References:

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