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Fundamental group of spaces of simple polygons. (English) Zbl 1471.55020

Summary: The space of shapes of \(n\)-gons with marked vertices can be identified with \(\mathbb{C} \mathbb{P}^{n-2}\). The space of shapes of \(n\)-gons without marked vertices is the quotient of \(\mathbb{C} \mathbb{P}^{n-2}\) by a cyclic group of order \(n\) generated by the function which re-enumerates the vertices. In this paper, we prove that the subset corresponding to simple polygons, i.e., without self-intersections, in each case is open and has two homeomorphic, simply connected components.

MSC:

55R80 Discriminantal varieties and configuration spaces in algebraic topology
52B99 Polytopes and polyhedra
51M05 Euclidean geometries (general) and generalizations
57M05 Fundamental group, presentations, free differential calculus
55Q52 Homotopy groups of special spaces
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