Surface group

In mathematics, the Seifert–van Kampen theorem of algebraic topology (named after Herbert Seifert and Egbert van Kampen), sometimes just called van Kampen's theorem, expresses the structure of the fundamental group of a topological space ${\displaystyle X}$ in terms of the fundamental groups of two open, path-connected subspaces that cover ${\displaystyle X}$. It can therefore be used for computations of the fundamental group of spaces that are constructed out of simpler ones.

Let X be a topological space which is the union of two open and path connected subspaces U₁, U₂. Suppose U₁ ∩ U₂ is path connected and nonempty, and let x₀ be a point in U₁ ∩ U₂ that will be used as the base of all fundamental groups. The inclusion maps of U₁ and U₂ into X induce group homomorphisms ${\displaystyle j_{1}:\pi _{1}(U_{1},x_{0})\to \pi _{1}(X,x_{0})}$ and ${\displaystyle j_{2}:\pi _{1}(U_{2},x_{0})\to \pi _{1}(X,x_{0})}$. Then X is path connected and ${\displaystyle j_{1}}$ and ${\displaystyle j_{2}}$ form a commutative pushout diagram:

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