Math4202 Topology II (Lecture 34)

Algebraic Topology

Fundamental group of surfaces

Definition of surface

A surface is a

• 2-dimensional manifold
• compact
  • Therefore, open disk, and $\mathbb{R}^2$ are not surfaces in our consideration.
• without boundary
  • Have no points with neighborhood that homeomorphic to upper half plane or lower half plane of $\mathbb{R}^2$.
  • Therefore, cylinder without cap is not in our consideration.
  • This is also called closed manifolds in some literature.

Decomposition of real projective space $\mathbb{R}P^2$

We cut the hemisphere cap and cut it.

Definition of polygonal region in the plane

Let $c\in \mathbb R$. Choose $a>0$ consider the circle in $\mathbb R^2$. centered at $c$ with radius $a>0$ Given a sequence of angles $\theta_0<\theta_1<\cdots<\theta_n=\theta_0+2\pi$, consider the points in the circle $p_i=c+a(\cos\theta_i,\sin\theta_i)$. We call $p_i$ the vertices of the polygonal region. Let $e_i$ be the edges between $p_i$ and $p_{(i+1)\mod n}$.

Note that since our surface is compact, the polygonal is always finite

Definition of polygonal region

Let $P$ be the polygonal region in the plane $A$. Labelling of the edges of $P$ is a map from the set of edges of $P$ to a set $S$ of labels. Given an orientation of each edge of $P$ and a labelling of the edges of $P$, we define an equivalence relation on $P$.

• Each point in the interior of $P$ is equivalent only to itself.
• Two edges with the same label. Let $h$ be a positive linear map from one edge to the other, preserving the orientation. Define $x$ in one edge to be equivalent to $h(x)$ in the other edge.

The quotient of $P$ by this equivalence relation is the surface obtained by pasting the edges together.