Theorem of van Kampen

In algebraic Topology, the theorem of van Kampen , also called theorem of Seifert-Van Kampen , is a result making it possible to calculate the fundamental Groupe of a topological Espace which breaks up into simpler spaces whose fundamental groups are already known.

Statement

Are $U\_1$, $U\_2$ the open related ones by arcs like their intersection, and is $x\; \backslash \; in\; U\_1\; \backslash \; course\; U\_2$. Then the fundamental group of $U\_1\; \backslash \; cup\; U\_2$ in $x$ is equal to the Produit fiber of the fundamental groups of $U\_1$ and $U\_2$ above that of $U\_1\; \backslash \; course\; U\_2$:

$\backslash \; pi\; (U\_1\; \backslash \; cup\; U\_2,\; X)\; =\; \backslash \; pi\; (U\_1,\; X)\; *\_\; \{\backslash \; pi\; (U\_1\; \backslash \; course\; U\_2,\; X)\}\; \backslash \; pi\; (U\_2,\; X)$.

An essential particular case is that where $U\_1\; \backslash \; course\; U\_2$ is simply related: $\backslash \; pi\; (U\_1\; \backslash \; cup\; U\_2,\; X)$ is then the free Produit $\backslash \; pi\; (U\_1,\; X)\; *\; \backslash \; pi\; (U\_2,\; X)$ of the fundamental groups of $U\_1$ and $U\_2$.

For example, a bored torus of a hole is homeomorphic with the meeting of two cylinders of simply related intersection. The theorem of van Kampen shows that its fundamental group is $\backslash \; mathbb\; Z^2$. In a similar way, the fundamental group of the projective Plan is the group with two elements.

Case of two closed subspaces

The theorem stated above remains valid if $U\_1$, $U\_2$ and $U\_1\; \backslash \; course\; U\_2$ are related subspaces closed by arcs.

Kampen

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