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
Full Text:

References:

 [1] M.A. Armstrong, The fundamental group of the orbit space of a discontinuous group, Math. Proc. Cambr. Philos. Soc. 64 (1968), 299–301. · Zbl 0159.53002 [2] C. Bavard and E. Ghys, Polygones du plan et polyÃ¨dres hyperboliques, Geom. Ded. 43 (1992), 207–224. · Zbl 0758.52001 [3] M.P. Do Carmo, Riemannian geometry, in Mathematics: Theory & applications, Birkhäuser, Berlin, 1992. [4] A. González, Topological properties of the spaces of simple and convex polygons up to orientation-preserving similarities, Ph.D. dissertation, CIMAT, Guanajuato, Mexico, 2014 (in Spanish). [5] A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. · Zbl 1044.55001 [6] M. Kapovich and J. Millson, On the moduli space of polygons in the Euclidean plane, J. Diff. Geom. 42 (1995), 133–164. · Zbl 0847.51026 [7] S. Kojima and Y. Yamashita, Shapes of stars, Proc. Amer. Math. Soc. 117 (1993), 845–851. · Zbl 0774.57011 [8] K.W. Kwun, Uniqueness of the open cone neighbourhoods, Proc. Amer. Math. Soc. 15 (1964), 476–479. · Zbl 0129.38204 [9] J.L. López-López, The area as a natural pseudo-Hermitian structure on the spaces of plane polygons and curves, Diff. Geom. Appl. 28 (2010), 582–592. · Zbl 1339.58003 [10] W. Thurston, Shapes of polyhedra and triangulations of the sphere, in The Epstein birthday schrift, Geom. Topol. Monogr. 1 (1998), 511–549. · Zbl 0931.57010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.