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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