# Math4201 Topology II (Lecture 11)

## Algebraic topology

### Fundamental group

The $*$ operation has the following properties:

#### Properties for the path product operation

Let $[f],[g]\in \Pi_1(X)$, for $[f]\in \Pi_1(X)$, let $s:\Pi_1(X)\to X, [f]\mapsto f(0)$ and $t:\Pi_1(X)\to X, [f]\mapsto f(1)$.

Note that $t([f])=s([g])$, $[f]*[g]=[f*g]\in \Pi_1(X)$.

This also satisfies the associativity. $([f]*[g])*[h]=[f]*([g]*[h])$.

We have left and right identity. $[f]*[e_{t(f)}]=[f], [e_{s(f)}]*[f]=[f]$.

We have inverse. $[f]*[\bar{x}]=[e_{s(f)}], [\bar{x}]*[f]=[e_{t(f)}]$

#### Definition for Groupoid

Let $f,g$ be paths where $g,f:[0,1]\to X$, and consider the function of all pathes in $G$, denoted as $\mathcal{G}$,

Set $t:\mathcal{G}\to X$ be the source map, for this case $t(f)=f(0)$, and $s:\mathcal{G}\to X$ be the target map, for this case $s(f)=f(1)$.

We define

$$
\mathcal{G}^{(2)}=\{(f,g)\in \mathcal{G}\times \mathcal{G}|t(f)=s(g)\}
$$

And we define the operation $*$ on $\mathcal{G}^{(2)}$ as the path product.

This satisfies the following properties:

- Associativity: $(f*g)*h=f*(g*h)$

Consider the function $\eta:X\to \mathcal{G}$, for this case $\eta(x)=e_{x}$.

- We have left and right identity: $\eta(t(f))*f=f, f*\eta(s(f))=f$

- Inverse: $\forall g\in \mathcal{G}, \exists g^{-1}\in \mathcal{G}, g*g^{-1}=\eta(s(g))$, $g^{-1}*g=\eta(t(g))$

#### Definition for loop

Let $x_0\in X$. A path starting and ending at $x_0$ is called a loop based at $x_0$.

#### Definition for the fundamental group

The fundamental group of $X$ at $x$ is defined to be

$$
(\Pi_1(X,x),*)
$$

where $*$ is the product operation, and $\Pi_1(X,x)$ is the set o homotopy classes of loops in $X$ based at $x$.

<details>
<summary>Example of fundamental group</summary>

Consider $X=[0,1]$, with subspace topology from standard topology in $\mathbb{R}$.

$\Pi_1(X,0)=\{e\}$, (constant function at $0$) since we can build homotopy for all loops based at $0$ as follows $H(s,t)=(1-t)f(s)+t$.

And $\Pi_1(X,1)=\{e\}$, (constant function at $1$.)

---

Let $X=\{1,2\}$ with discrete topology.

$\Pi_1(X,1)=\{e\}$, (constant function at $1$.)

$\Pi_1(X,2)=\{e\}$, (constant function at $2$.)

---

Let $X=S^1$ be the circle.

$\Pi_1(X,1)=\mathbb{Z}$ (related to winding numbers, prove next week).

</details>

A natural question is, will the fundamental group depends on the base point $x$?

#### Definition for $\hat{\alpha}$

Let $\alpha$ be a path in $X$ from $x_0$ to $x_1$. $\alpha:[0,1]\to X$ such that $\alpha(0)=x_0$ and $\alpha(1)=x_1$. Define $\hat{\alpha}:\Pi_1(X,x_0)\to \Pi_1(X,x_1)$ as follows:

$$
\hat{\alpha}(\beta)=[\bar{\alpha}]*[f]*[\alpha]
$$

#### $\hat{\alpha}$ is a group homomorphism

$\hat{\alpha}$ is a group homomorphism between $(\Pi_1(X,x_0),*)$ and $(\Pi_1(X,x_1),*)$

<details>
<summary>Proof</summary>

Let $f,g\in \Pi_1(X,x_0)$, then $\hat{\alpha}(f*g)=\hat{\alpha}(f)\hat{\alpha}(g)$

$$
\begin{aligned}
\hat{\alpha}(f*g)&=[\bar{\alpha}]*[f]*[g]*[\alpha]\\
&=[\bar{\alpha}]*[f]*[e_{x_0}]*[g]*[\alpha]\\
&=[\bar{\alpha}]*[f]*[\alpha]*[\bar{\alpha}]*[g]*[\alpha]\\
&=([\bar{\alpha}]*[f]*[\alpha])*([\bar{\alpha}]*[g]*[\alpha])\\
&=(\hat{\alpha}(f))*(\hat{\alpha}(g))
\end{aligned}
$$

---

Next, we will show that $\hat{\alpha}\circ \hat{\bar{\alpha}}([f])=[f]$, and $\hat{\bar{\alpha}}\circ \hat{\alpha}([f])=[f]$.

$$
\begin{aligned}
\hat{\alpha}\circ \hat{\bar{\alpha}}([f])&=\hat{\alpha}([\bar{\alpha}]*[f]*[\alpha])\\
&=[\alpha]*[\bar{\alpha}]*[f]*[\alpha]*[\bar{\alpha}]\\
&=[e_{x_0}]*[f]*[e_{x_1}]\\
&=[f]
\end{aligned}
$$

The other case is the same

</details>

#### Corollary of fundamental group

If $X$ is path-connected and $x_0,x_1\in X$, then $\Pi_1(X,x_0)$ is isomorphic to $\Pi_1(X,x_1)$.
