Seifert–van Kampen 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. He makes a statement about the structure of the fundamental group of a topological space X, by the fundamental groups of two open, path-connected subspaces U and V, which cover X is considered. So you can calculate the fundamental group of complicated spaces from those of simpler spaces.

The easy half of the sentence

It is a path-connected space dotted. Next is an open cover of X by path-connected subsets that contain * all the point pairs and their sections are each also path-connected.

For his inclusion. Then generated from the sub-groups

The statement is therefore that the relative homotopy classes of closed paths in X, the whole run in one, generate the fundamental group of X. In particular, X is simply connected if each has this property.

The actual set of Seifert and van Kampen

Let X be a path-connected topological space, open and path-connected, so true, and. It should also be path-connected. Among the inclusions from to include (not necessarily injective ) homomorphisms

Among the inclusions of X are homomorphisms by

Obviously applies This further be any group H, and having the property of group homomorphisms

Then there exists a unique group homomorphism, so

So the set of Seifert and van Kampen says a universal imaging property of the first fundamental group.

Combinatorial version

In the language of combinatorial group theory is the amalgamated product of and via the homomorphisms and. If these three fundamental groups Präsentierungen following:

Then the amalgamation can be used as

Are presented. The fundamental group of is thus generated by the loops in the subspaces and; additional relations is added only that a loop in the section regardless of whether one perceives it as an element of or represents the same element.

Example for Lemma

Take from the n- dimensional sphere and Q, P be two distinct points. Then and path-connected. Your average is path due also.

But now, by means of the stereographic projection, homeomorphic to. Since is contractible, so this also applies to and therefore will have this trivial fundamental groups. This is not from the base point dependent. Therefore, it is also trivial.

Conclusions

If the fundamental group is trivial, then says the set of Seifert and van Kampen, that the free product of and is. It is produced by these groups and between producers, there are no relations, which would not have been in or out. In particular, and injective.