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Section 2.8 Fundamental group of \(S^n\)
Theorem 2.8.1 . Weak Seifert-van Kampen.
Let \(X=U\cup V\) be an open covering, and suppose \(U\cap V\) is path connected. Given any \(x_0\in U\cap V\text{,}\) let
\begin{align*}
i_*\colon \pi_1(U,x_0)\amp \rightarrow \pi_1(X, x_0) \amp j_*\colon \pi_1(V,x_0)\amp \rightarrow \pi_1(X, x_0)
\end{align*}
be the group homomorphisms corresponding to the inclusion maps \(i\colon U\hookrightarrow X\text{,}\) \(j\colon V\hookrightarrow X\text{.}\) The group \(\pi_1(X,x_0)\) is generated by the images \(i_*(\pi_1(U,x_0))\) and \(j_*(\pi_1(V,x_0))\text{.}\) In other words, for all \([f]\in \pi_1(X,x_0)\text{,}\) we have
\begin{equation*}
[f]=[g_1]*[g_2]*\cdots *[g_n]
\end{equation*}
where for all \(1\leq k\leq n\) we have \([g_k]\in i_*(\pi_1(U,x_0))\) or \([g_k]\in j_*(\pi_1(V,x_0))\text{.}\)
Corollary 2.8.2 . Fundamental group of \(S^n\) .
For any \(n\geq 1\) the \(n\) -sphere\(S^n\) is defined as
\begin{equation*}
S^n=\{\boldx\in \R^{n+1}\colon \norm{\boldx}=1\}\subseteq \R^{n+1}.
\end{equation*}
When \(n=1\text{,}\) we have \(\pi_1(S^1, \boldx_0)=\Z\) for any \(\boldx_0\in S^1\text{.}\)
For all \(n\geq 2\) we have \(\pi_1(S^n, \boldx_0)=\{e\}\) for any \(\boldx_0\in S^n\text{.}\)
Since \(S^n\) is path connected for all \(n\geq 1\text{,}\) we conclude that \(S^n\) is simply connected for all \(n\geq 2\text{.}\)
Proof. The alternate description of $S^n$ was shown in a homework exercise.
Definition 2.8.3 . Projective space.
Fix \(n\geq 1\text{.}\) (Real) projective \(n\) -space \(\PP^n\) is the quotient space of \(\R^{n+1}-\{\boldzero\}\) by the equivalence relation \(\boldx\sim \boldy \iff \boldx=c\boldy\) for some \(c\in \R\text{.}\)
As an alternative description, we have \(\PP^n\cong S^{n}/\sim\text{,}\) where \(\sim\) is the antipodal relation: i.e., for all \(\boldx, \boldy\in S^n\) we define \(\boldx\sim \boldy\) if and only if \(\boldx=\pm \boldy\text{.}\)
Corollary 2.8.4 . Fundamental group of \(\PP^n\) .
Fix \(n\geq 2\text{.}\)
The quotient map \(q\colon S^n\rightarrow \PP^n\) is a double covering of \(\PP^n\text{.}\)
For all \(\boldx_0\in \PP^n\text{,}\) we have \(\pi_1(\PP^n, \boldx_0)=\Z/2\Z\text{.}\)
