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 $S$ .)

Proof

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}

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