Calculating the fundamental group

There are several tools that facilitate calculating the fundamental group. Van Kampen’s theorem can be used to compute the fundamental group of a space in terms of simpler spaces it is constructed from. If certain conditions are met, the theorem states that for \({X=\bigcup A_{\alpha}}\), \({\pi_{1}\left(X\right)=\underset{\alpha}{*}\pi_{1}\left(A_{\alpha}\right)}\), the free product of the component fundamental groups. Under less restrictive conditions this becomes a factor group of the free product. For example the result for the figure eight \({S^{1}\vee S^{1}}\) can be generalized to the statement that the fundamental group of any wedge product is the free product of the fundamental groups of its constituents.

We also have the fact that for path-connected \({X}\) and \({Y}\), \({\pi_{1}\left(X\times Y\right)=\pi_{1}\left(X\right)\times\pi_{1}\left(Y\right)}\), so that for the torus we have \({\pi_{1}\left(T^{2}\right)=\pi_{1}\left(S^{1}\times S^{1}\right)=\pi_{1}\left(S^{1}\right)\times\pi_{1}\left(S^{1}\right)=\mathbb{Z}\times \mathbb{Z}}\), a case in which the fundamental group is abelian. For path-connected spaces \({X}\), \({H_{1}(X)}\) is the abelianization of \({\pi_{1}\left(X\right)}\), as for example is the case for the figure eight. In particular, \({H_{1}(X)}\) and \({\pi_{1}\left(X\right)}\) are identical for \({S^{n}}\), \({T^{n}}\), and \({\mathbb{C}\textrm{P}^{n}}\).

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