Rocky Mountain Journal of Mathematics

Fundamental group of spaces of simple polygons

Ahtziri Gonzalez

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

Rocky Mountain J. Math., Volume 48, Number 6 (2018), 1871-1886.

First available in Project Euclid: 24 November 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 54B05: Subspaces 55Q52: Homotopy groups of special spaces

Space of polygons simple polygons polygons without marked vertices


Gonzalez, Ahtziri. Fundamental group of spaces of simple polygons. Rocky Mountain J. Math. 48 (2018), no. 6, 1871--1886. doi:10.1216/RMJ-2018-48-6-1871.

Export citation


  • M.A. Armstrong, The fundamental group of the orbit space of a discontinuous group, Math. Proc. Cambr. Philos. Soc. 64 (1968), 299–301.
  • C. Bavard and E. Ghys, Polygones du plan et polyèdres hyperboliques, Geom. Ded. 43 (1992), 207–224.
  • M.P. Do Carmo, Riemannian geometry, in Mathematics: Theory & applications, Birkhäuser, Berlin, 1992.
  • 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).
  • A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002.
  • M. Kapovich and J. Millson, On the moduli space of polygons in the Euclidean plane, J. Diff. Geom. 42 (1995), 133–164.
  • S. Kojima and Y. Yamashita, Shapes of stars, Proc. Amer. Math. Soc. 117 (1993), 845–851.
  • K.W. Kwun, Uniqueness of the open cone neighbourhoods, Proc. Amer. Math. Soc. 15 (1964), 476–479.
  • 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.
  • W. Thurston, Shapes of polyhedra and triangulations of the sphere, in The Epstein birthday schrift, Geom. Topol. Monogr. 1 (1998), 511–549.