- State the Seifert–van Kampen Theorem.
- Calculate the fundamental group of the space obtained by taking two copies of the torus \(T = S^1 \times S^1\) and gluing them along a circle \(S^1 \times \{p\}\).
- Calculate the fundamental group of the Klein bottle \(K\).
- Calculate the fundamental group of the wedge sum \(T^2 \vee S^1 = (S^1 \times S^1) \vee S^1\).
- Calculate the fundamental group of the wedge sum \(T^2 \vee \mathbb{RP}^2\).
Solution
1. Seifert–van Kampen. If \(X=U\cup V\), where \(U,V,U\cap V\) are path-connected open sets containing a common basepoint \(x_0\), then\[ \pi_1(X,x_0) \cong \pi_1(U,x_0)*_{\pi_1(U\cap V,x_0)}\pi_1(V,x_0), \]where the amalgamation uses the two inclusion-induced homomorphisms.
2. For two tori glued along a circle representing one standard generator in each torus,\[ \pi_1 \cong \langle a,b_1,b_2\mid[a,b_1]=1,\ [a,b_2]=1\rangle. \]Since \(a\) commutes with both free generators \(b_1,b_2\), this is\[ \mathbb Z\times F_2. \]
3. The Klein bottle has the standard presentation\[ \pi_1(K) \cong \langle a,b\mid aba^{-1}=b^{-1}\rangle \cong \mathbb Z\rtimes_{-1}\mathbb Z, \]where the generator represented by \(a\) acts on \(\langle b\rangle\cong\mathbb Z\) by inversion.
4. For a wedge of connected CW complexes, van Kampen gives a free product. Hence\[ \pi_1(T^2\vee S^1) \cong \mathbb Z^2*\mathbb Z \cong \langle a,b,c\mid[a,b]=1\rangle. \]
5. Likewise,\[ \pi_1(T^2\vee\mathbb{RP}^2) \cong \mathbb Z^2*(\mathbb Z/2) \cong \langle a,b,d\mid[a,b]=1,\ d^2=1\rangle. \]