# Math4202 Topology II (Lecture 27)

## Algebraic Topology

### Fundamental Groups for Higher Dimensional Sphere

#### Theorem for "gluing" fundamental group

Suppose $X=U\cup V$, where $U$ and $V$ are open subsets of $X$. Suppose that $U\cap V$ is path connected, and $x\in U\cap V$. Let $i,j$ be the inclusion maps of $U$ and $V$ into $X$, the images of the induced homomorphisms

$$
i_*:\pi_1(U,x_0)\to \pi_1(X,x_0)\quad j_*:\pi_1(V,x_0)\to \pi_1(X,x_0)
$$

The image of the two map generate $\pi_1(X,x_0)$.

$G$ is a group, and let $S\subseteq G$, where $G$ is generated by $S$, if $\forall g\in G$, $\exists s_1,s_2,\ldots,s_n\in S$ such that $g=s_1s_2\ldots s_n\in G$. (We can write $G$ as a word of elements in $S$.)

<details>
<summary>Proof</summary>

Let $f$ be a loop in $X$, $f\simeq g_1*g_2*\ldots*g_n$, where $g_i$ is a loop in $U$ or $V$.

For example, consider the function, $f=f_1*f_2*f_3*f_4$, where $f_1\in S_+$, $f_2\in S_-$, $f_3\in S_+$, $f_4\in S_-$.

Take the functions $\bar{\alpha_1}*\alpha_1\simeq e_{x_1}$ where $x_1$ is the intersecting point on $f_1$ and $f_2$.

Therefore,

$$
\begin{aligned}
f&=f_1*f_2*f_3*f_4\\
&(f_1*\bar{\alpha})*(\alpha_1*f_2*\bar{\alpha_2})*(\alpha_2*f_3*\bar{\alpha_3})*(\alpha_4*f_4)
\end{aligned}
$$

This decompose $f$ into a word of elements in either $S_+$ or $S_-$.

---

Note that $f$ is a continuous function $I\to X$, for $t\in I$, $\exists I_t$ being a small neighborhood of $t$ such that $f(I_t)\subseteq U$ or $f(I_t)\subseteq V$.

Since $U_{t\in I}I_t=I$, then $\{I_t\}_{t\in I}$ is an open cover of $I$.

By compactness of $I$, there is a finite subcover $\{I_{t_1},\ldots,I_{t_n}\}$.

Therefore, we can create a partition of $I$ into $[s_i,s_{i+1}]\subseteq I_{t_k}$ for some $k$.

Then with the definition of $I_{t_k}$, $f([s_i,s_{i+1}])\subseteq U$ or $V$.

Then we can connect $x_0$ to $f(s_i)$ with a path $\alpha_i\subseteq U\cap V$.

$$
\begin{aligned}
f&=f|_{[s_0,s_1]}*f|_{[s_1,s_2]}*\ldots**f|_{[s_{n-1},s_n]}\\
&\simeq f|_{[s_0,s_1]}*(\bar{\alpha_1}*\alpha_1)*f|_{[s_1,s_2]}*(\bar{\alpha_2}*\alpha_2)*\ldots*f|_{[s_{n-1},s_n]}*(\bar{\alpha_n}*\alpha_n
)\\
&=(f|_{[s_0,s_1]}*\bar{\alpha_1})*(\alpha_1*f|_{[s_1,s_2]}*\bar{\alpha_2})*\ldots*(\alpha_{n-1}*f|_{[s_{n-1},s_n]}*\bar{\alpha_n})\\
&=g_1*g_2*\ldots*g_n
\end{aligned}
$$

</details>

#### Corollary in higher dimensional sphere

Since $S^n_+$ and $S^n_-$ are homeomorphic to open balls $B^n$, then $\pi_1(S^n_+,x_0)=\pi_1(S^n_-,x_0)=\pi_1(B^n,x_0)=\{e\}$ for $n\geq 2$.

> Preview: Van Kampen Theorem