Pasted polygonal regions

Section 2.19 Pasted polygonal regions

Definition 2.19.1. Polygonal region.

Let $C\subseteq \R^2$ be the the circle with equation $(x-a)^2+(y-b)^2=R^2\text{.}$ Given integer $n\geq 3$ and angles

\begin{equation*} \theta_0 < \theta_1 < \dots < \theta_n=\theta_0+2\pi \end{equation*}

we consider the $n$ distinct points on $C$ given by

\begin{equation*} p_k=(a,b)+R(\cos\theta_k, \sin\theta_k)\text{,} \end{equation*}

for $0\leq k\leq n\text{.}$

Next for each $1\leq k\leq n\text{,}$ let $H_k$ be the closed half-plane containing $\{p_k\}_{k=1}^n$ whose boundary is the line $\ell$ determined by $\{p_{k-1}, p_{k}\}\text{.}$ The polygonal region determined by $p_1,p_2,\dots, p_n=p_0$ is the region $P$ defined as

\begin{equation*} P=H_1\cap H_2\cap\cdots \cap H_n\text{.} \end{equation*}

For all $1\leq k\leq n$ the point $p_k$ is called a vertex of $P\text{,}$ and the line segment between $p_{k-1}$ and $p_{k}$ is called an edge of $P\text{.}$

Definition 2.19.2. Oriented line segment.

An orientation of a line segment $L\subseteq \R^2$ is an ordering of its endpoints as a pair $(p,q)\text{.}$ In this case $p$ is called the initial point, and $q$ the final point of $L\text{.}$ We will denote by $L_{p,q}$ the oriented line segment with initial point $p$ and final point $q\text{.}$

Given two oriented line segments $L_{p,q}$ and $L_{p',q'}$ the positive linear map of $L_{p,q}$ onto $L_{p',q'}$ is the unique homeomorphism $h$ satisfying

\begin{equation*} h((1-t)p+tq)=(1-t)p'+tq' \end{equation*}

for all $t\in [0,1]\text{.}$

Remark 2.19.3.

If $P$ is the polygonal region corresponding to points $p_1, p_2,\dots, p_n=p_0\text{,}$ and $P'$ is the polygonal region corresponding to points $p_1',p_2',\dots, p_n'=p_0'\text{,}$ then there is a homeomorphism $h\rightarrow P\rightarrow P'$ whose restriction to the oriented edge $L_{p_{k-1},p_{k}}$ is the positive linear map from $L_{p_{k-1},p_k}$ to $L_{p_{k-1}',p_k'}\text{.}$

Definition 2.19.4. Oriented labelling.

Let $P$ be a polygonal region corresponding to points $p_1,p_2,\dots, p_n=p_0\text{,}$ for each $1\leq k\leq n$ let $E_k$ be the edge between $p_{k-1}$ and $p_k\text{,}$ and let $E=\{E_1,E_2,\dots, E_n\}$ be the set of edges of $P\text{.}$ A labelling of the edges of $P$ is a function

\begin{equation*} \ell\colon E\rightarrow S \text{.} \end{equation*}

For each $E_k\in E\text{,}$ we call $\ell(E_k)\in S$ the label of $E_k\text{.}$

An oriented labelling of the edges of $P$ is a function

\begin{equation*} \ell\colon E\rightarrow S\times \{1,-1\}\text{.} \end{equation*}

For each $E_k\in E$ the assigned orientation of $E_k$ with respect to $\ell$ is $(p_{k-1}, p_k)$ if $\ell(E_k)=(s,1)\text{,}$ and $(p_k,p_{k-1})$ if $\ell(E_k)=(s,-1)\text{.}$

Given an oriented labelling $l$ of $P\text{,}$ the corresponding labelling scheme of length $n$ is the expression

\begin{equation*} s_1^{\epsilon_1} s_2^{\epsilon_2}\cdots s_n^{\epsilon_n}\text{,} \end{equation*}

where for all $1\leq k\leq n$ we have $\ell(E_k)=(s_k, \epsilon_k)\text{.}$ In other words, $s_k$ is the label of edge $E_k\text{,}$ and $\epsilon_k$ indicates its orientation.

Definition 2.19.5. Pasted polygonal region.

Let $P$ be a polygonal region corresponding to points $p_1,p_2,\dots, p_n=p_0$ and let

\begin{equation*} w=s_1^{\epsilon_1} s_2^{\epsilon_2}\cdots s_n^{\epsilon_n} \end{equation*}

be a labelling scheme corresponding to an oriented labelling of the edges of $P\text{.}$ The space $X$ obtained by pasting the edges of $P$ together according to the labelling scheme $w$ is the quotient obtained by identifying points on any two oriented edges $E_j=L_{p,q}$ and $E_{k}=L_{p',q'}$ that have the same label according to the positive linear map $h\colon L_{p,q}\rightarrow L_{p',q'}\text{.}$

More generally, given pairwise disjoint polygonal regions $P_1,P_2,\dots, P_r$ with labelling schemes $w_1,w_2,\dots, w_n\text{,}$ the space obtained by pasting together the edges of the $P_k$ is the quotient obtained from $\bigcup_{k=1}^r P_k$ by identifying points on edges with the same labels as above.

Theorem 2.19.6. Pasted polygonal regions.

If $X$ is obtained by pasting together the edges of polygonal regions according to labelling schemes, then $X$ is a compact Hausdorff space.

Theorem 2.19.7. Pasted polygonal fundamental groups.

Let $X$ be obtained by pasting together the edges of a polygon $P$ according to the labelling scheme

\begin{equation*} w=a_{i_1}^{\epsilon_1} a_{i_2}^{\epsilon_2}\cdots a_{i_n}^{\epsilon_n}\text{,} \end{equation*}

let $\{a_1,a_2,\dots, a_m\}$ be the distinct labels occurring in $w\text{,}$ and let $q\colon P\rightarrow X $ be the quotient map. If $q$ maps all vertices of $P$ to a single point $x_0\in X\text{,}$ then

\begin{equation*} \pi_1(X,x_0)=\langle a_1,a_2,\dots, a_m\vert a_{i_1}^{\epsilon_1} a_{i_2}^{\epsilon_2}\cdots a_{i_n}^{\epsilon_n}\rangle\text{.} \end{equation*}

Definition 2.19.8. $n$-fold torus.

Fix an integer $n\geq 1\text{.}$ Let $\{a_1,a_2,\dots, a_n\}$ and $\{b_1,b_2,\dots, b_n\}$ be disjoint sets of cardinality $n\text{,}$ and let $X$ be the space obtained by pasting together the edges of a polygon according to the labelling scheme

\begin{equation*} w=a_1b_1a_1^{-1}b_{1}^{-1}a_2b_2a_2^{-1}b_2^{-1}\cdots a_nb_na_{n}^{-1}b_{n}^{-1}\text{.} \end{equation*}

We call $X$ an $n$-fold connected sum of tori, or simply the $n$-fold torus, denoted $T\# T\# \cdots \# T\text{.}$

Definition 2.19.9. $m$-fold projective plane.

Fix an integer $m\geq 2\text{.}$ Let $a_1,a_2,\dots, a_m$ be distinct labels, and let $X$ be the space obtained by pasting together the edges of a polygon according to the labelling scheme

\begin{equation*} w=a_1a_1a_2a_2\cdots a_ma_m\text{.} \end{equation*}

We call $X$ an $m$-fold connected sum of projective planes, or simply the $m$-fold projective plane, denoted $P^2\# P^2\cdots P^2\text{.}$

Corollary 2.19.10. Fundamental group of $n$-fold torus and $m$-fold projective plane.

Let $X$ be the space obtained by pasting together the edges of a polygon $P$ according to the labelling scheme $w\text{,}$ and suppose all vertices get mapped to a single point $x_0\in X\text{.}$

$n$-fold torus.

If $X$ is an $n$-fold torus, then

\begin{equation*} \pi_1(X,x_0)=\langle a_1,b_1,a_2,b_2,\dots , a_n,b_n\vert [a_1,b_1][a_2,b_2]\cdots [a_n,b_n]\rangle\text{.} \end{equation*}
$m$-fold projective plane.

If $X$ is an $m$-fold projective plane, then

\begin{equation*} \pi_1(X,x_0)=\langle a_1,a_2,\dots , a_m\vert a_1^2a_2^2\cdots a_m^2\rangle\text{.} \end{equation*}

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