Seifert and van Kampen's theorem

The set of Seifert and van Kampen (named after Herbert Seifert and Egbert van Kampen ) is a mathematical theorem in the field of algebraic topology . It makes a statement about the structure of the fundamental group of a topological space X by considering the fundamental groups of two open, path-connected subspaces U and V, which cover X. So one can calculate the fundamental group of complicated spaces from those of simpler spaces.

The easy half of the sentence

It is a dotted space connected by a path . Furthermore, let X be an open cover by path-connected subsets, which all contain the point * and whose pairwise intersections are also path-connected. ${\displaystyle (X,*)}$ ${\displaystyle (U_{\lambda })_{\lambda \in \Lambda }}$

For be the inclusion. Then the subgroups are generated ${\displaystyle \lambda \in \Lambda }$ ${\displaystyle f_{\lambda }:(U_{\lambda },*)\rightarrow (X,*)}$ ${\displaystyle \pi _{1}(X,*)}$ ${\displaystyle \pi _{1}(f_{\lambda })(U_{\lambda },*),\lambda \in \Lambda .}$

So the statement is that the relative homotopy classes in X of closed paths that run entirely in one generate the fundamental group of X. In particular, X is simply connected if each has this property. ${\displaystyle U_{\lambda }}$ ${\displaystyle U_{\lambda }}$

The actual sentence by Seifert and van Kampen

Let it be a path-connected topological space, open and path-connected, so that , and . Also be path-related. The inclusions from to include (not necessarily injective ) homomorphisms ${\displaystyle X}$ ${\displaystyle U_{1},U_{2}\subseteq X}$ ${\displaystyle X=U_{1}\cup U_{2}}$ ${\displaystyle *\in U_{3}:=U_{1}\cap U_{2}}$ ${\displaystyle U_{3}}$ ${\displaystyle U_{3}}$ ${\displaystyle U_{1},U_{2}}$

${\displaystyle v_{i}:\pi _{1}(U_{3},*)\rightarrow \pi _{1}(U_{i},*),i=1,2.}$

The inclusions from to include homomorphisms ${\displaystyle U_{j}}$ ${\displaystyle X}$

${\displaystyle u_{j}:\pi _{1}(U_{j},*)\rightarrow \pi _{1}(X,*),1\leq j\leq 3.}$

Obviously, let H be an arbitrary group and group homomorphisms with the property ${\displaystyle u_{3}=u_{i}\circ v_{i},i=1,2.}$ ${\displaystyle p_{j}:\pi _{1}(U_{j},*)\rightarrow H}$

${\displaystyle p_{3}=p_{i}\circ v_{i},i=1,2.}$

Then there is a uniquely determined group homomorphism such that ${\displaystyle p:\pi _{1}(X,*)\rightarrow H}$

${\displaystyle p_{j}=p\circ u_{j},1\leq j\leq 3.}$

So the Seifert and van Kampen theorem states a universal mapping property of the first fundamental group.

Combinatorial version

In the language of combinatorial group theory , the amalgamated product of and over is via the homomorphisms and . When these three fundamental groups have the following presentations : ${\displaystyle \pi _{1}(X,*)}$ ${\displaystyle \pi _{1}(U_{1},*)}$ ${\displaystyle \pi _{1}(U_{2},*)}$ ${\displaystyle \pi _{1}(U_{3},*)}$ ${\displaystyle u_{1}}$ ${\displaystyle u_{2}}$

${\displaystyle \pi _{1}(U_{1},*)=\langle \alpha _{1},...,\alpha _{k}|r_{1},...,r_{l}\rangle }$,

${\displaystyle \pi _{1}(U_{2},*)=\langle \beta _{1},...,\beta _{m}|s_{1},...,s_{n}\rangle }$ and

${\displaystyle \pi _{1}(U_{3},*)=\langle \gamma _{1},...,\gamma _{p}|t_{1},...,t_{q}\rangle }$,

then the amalgamation as

${\displaystyle \pi _{1}(X,*)=\pi _{1}(U_{1},*)\;*_{\pi _{1}(U_{3},*)}\;\pi _{1}(U_{2},*)}$

${\displaystyle =\langle \alpha _{1},...,\alpha _{k},\beta _{1},...,\beta _{m}|r_{1},...,r_{l},s_{1},...,s_{n},v_{1}(\gamma _{1})=v_{2}(\gamma _{1}),...,v_{1}(\gamma _{p})=v_{2}(\gamma _{p})\rangle }$

to get presented. The fundamental group of is thus generated by the loops in the subspaces and ; the only additional relation is that a loop represents the same element in the section regardless of whether it is understood as an element of or of . ${\displaystyle X}$ ${\displaystyle U_{1}}$ ${\displaystyle U_{2}}$ ${\displaystyle U_{3}}$ ${\displaystyle \pi _{1}(U_{1},*)}$ ${\displaystyle \pi _{1}(U_{2},*)}$

Example for the auxiliary sentence

Take out the n-dimensional sphere and two different points . Then and are away related. Their average is because of also route related. ${\displaystyle S^{n},n\geq 2}$ ${\displaystyle P,Q}$ ${\displaystyle S^{n}}$ ${\displaystyle U_{1}:=S^{n}\setminus \{P\}}$ ${\displaystyle U_{2}:=S^{n}\setminus \{Q\}}$ ${\displaystyle n\geq 2}$

But now , by means of the stereographic projection, it is homeomorphic to . Since is contractible , this also applies to and and therefore they have trivial fundamental groups. This is not dependent on the base point. Hence is also trivial. ${\displaystyle S^{n}\setminus \{P\}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle U_{1}}$ ${\displaystyle U_{2}}$ ${\displaystyle \pi _{1}(S^{n})}$

Inferences

If the fundamental group is trivial, then Seifert and van Kampen's theorem says that the free product of and is. It is created by these groups and there are no relations between the producers that have not already been in or . In particular, and are injective. ${\displaystyle \pi _{1}(U_{3},*)}$ ${\displaystyle \pi _{1}(X,*)}$ ${\displaystyle \pi _{1}(U_{1},*)}$ ${\displaystyle \pi _{1}(U_{2},*)}$ ${\displaystyle \pi _{1}(U_{1},*)}$ ${\displaystyle \pi _{1}(U_{2},*)}$ ${\displaystyle u_{1}}$ ${\displaystyle u_{2}}$

See also

• Stefan Kühnlein, script: Introduction to topology (2008)
• Sebastian Hage, seminar lecture: Seifert-van Kampen's theorem - Gruppoide, Pushouts & Brown's theorem (2004; PDF; 517 kB). Contains a category-theoretical and a topological proof of the theorem.