## Rocky Mountain Journal of Mathematics

### Fundamental group of spaces of simple polygons

Ahtziri Gonzalez

#### Abstract

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

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

Dates
First available in Project Euclid: 24 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1543028442

Digital Object Identifier
doi:10.1216/RMJ-2018-48-6-1871

Mathematical Reviews number (MathSciNet)
MR3879306

Zentralblatt MATH identifier
06987229

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

#### Citation

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. https://projecteuclid.org/euclid.rmjm/1543028442

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