Seifert–van_kampen_theorem

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

The underlying idea is that paths in X can be partitioned: into journeys through the intersection W of U and V, through U but outside V, and through V outside U. In order to move segments of paths around, by homotopy to form loops returning to a base point w in W, we should assume U, V and W are path-connected; and that W isn\'t empty. We assume also that U and V are open subspaces with union X.

Under these conditions, π₁(U,w), π₁(V,w), and π₁(W,w), together with the inclusion homomorphisms (induced by the inclusion map):

I : π₁(W,w) → π₁(U,w)

and

J : π₁(W,w) → π₁(V,w),

are sufficient data to determine π₁(X,w). The maps I and J extend to an epimorphism

Φ : π₁(U,w) * π₁(V,w) → π₁(X,w),

where π₁(U,w) * π₁(V,w) is the free product of π₁(U,w) and π₁(V,w). The kernel of the map Φ are the loops in W that, when viewed in X, are homotopic to the trivial one at w. The group π₁(X,w) is therefore isomorphic to π₁(U,w) * π₁(V,w) modulo such elements.

In particular, when W is simply connected (so that its fundamental group is the trivial group), the theorem says that π₁(X,w) is isomorphic to the free product π₁(U,w)$*$π₁(V,w).

Equivalent formulations

In the language of combinatorial group theory, π₁(X,w) is the free product with amalgamation of those of U and V, with respect to the homomorphisms I and J (which might not be injective): given group presentations

π₁(U,w) = 1,...,u_k | α₁,...,α_l>

π₁(V,w) = 1,...,v_m | β₁,...,β_n>

π₁(W,w) = 1,...,w_p | γ₁,...,γ_q>

the amalgamation can be written in terms of generators and relations as π₁(X,w) = r)·J(w_r)^-1> where each letter u, v, w, α, β, γ stands for the respective set of generators or relators, and the final relator means that the images of each generator w_r under the inclusions I, J are equivalent in the fundamental group of X.

In category theory, the fundamental group of X is a colimit of the diagram of those of U, V and W. More precisely, π₁(X,w) is the pushout of the diagram.

Generalizations

This theorem has been extended to the non-connected case by using the fundamental groupoid π₁(X,A) on a set A of base points, which consists of homotopy classes of paths in X joining points of X which lie in A. The connectivity conditions for the theorem then become that A meets each path-component of U,V,W. The pushout is now in the category of groupoids. This extended theorem allows the determination of the fundamental group of the circle, and many other useful cases. Applications of the fundamental groupoid to the Jordan Curve theorem, covering spaces, and orbit spaces are given in Ronald Brown\'s book cited below.

There is also a version that allows more than two overlapping sets; for more information on this, see Allen Hatcher\'s book below, theorem 1.20.

In fact, we can extend van Kampen\'s theorem significantly farther by considering the fundamental groupoid $\backslash Pi(X)$, an element of the category of small categories whose objects are points of X and whose arrows are paths between points modulo homotopy equivalence. In this case, to determine the fundamental groupoid of a space, we need only know the fundamental groupoids of the open sets covering X as follows: create a new category in which the objects are fundamental groupoids of the open sets, with an arrow between groupoids if the domain space is a subspace of the codomain. Then van Kampen\'s theorem is the assertion that the fundamental groupoid of X is the colimit of the diagram. For details, see Peter May\'s book, chapter 2.

References

• Allen Hatcher, Algebraic topology. (2002) Cambridge University Press, Cambridge, xii+544 pp. ISBN 052179160X and ISBN 0521795400
• Peter May, A Concise Course in Algebraic Topology. (1999) University of Chicago Press, ISBN 0-226-51183-9. (Section 2.7 provides a category-theoretic presentation of the theorem as a colimit in the category of groupoids)
• Ronald Brown, Topology and groupoids (2006) Booksurge LLC ISBN 1-4196-2722-8
• P.J. Higgins, Categories and groupoids (1971) Van Nostrand Reinhold
• Ronald Brown, Higher dimensional group theory (2007) (Gives a broad view of higher dimensional van Kampen theorems involving multiple groupoids.)
• Van Kampen\'s theorem on PlanetMath
• Van Kampen\'s theorem result on PlanetMath

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