Introduction

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In mathematics, the Seifert–van Kampen theorem of algebraic topology, sometimes just called Van Kampen's theorem, explains the structure of the fundamental group of a topological space X, in terms of those of two overlapping subspaces U and V, under certain hypothesis about connectedness. It can therefore be used for computations of the fundamental group of spaces that are constructed out of simpler ones. It expresses the idea that paths in X can be separated out: 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.

The conditions are then enough to ensure that π₁(U,w), π₁(V,w), and π₁(W,w), together with the inclusion homomorphisms

I : π₁(W,w) → π₁(U,w)

and

J : π₁(W,w) → π₁(V,w),

are sufficient data to determine π₁(X,w). It is easier to state the result in case W is simply connected, so that its fundamental group is {e}. In that case the theorem says simply that the fundamental group of X is the free product of those of U and V.

Equivalent Statements

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In the general case (that is, dropping the assumption that W is simply connected) the fundamental group of X is a colimit of the diagram of those of U, V and W. In group theorists' terms, it 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) = <u₁,...,u_k | α₁,...,α_l>

π₁(V,w) = <v₁,...,v_m | β₁,...,β_n>

π₁(W,w) = <w₁,...,w_p | γ₁,...,γ_q>

the amalgamation can be written in terms of generators and relations as π₁(X,w) = <u, v, w | α, β, γ, I(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. Finally, in category theorists' terms, π₁(X,w) is the pushout of the diagram mentioned above.

Generalizations

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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.

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.

Reference

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Hatcher, Allen, Algebraic topology. Cambridge University Press, Cambridge, 2002. xii+544 pp. ISBN 052179160X and ISBN 0521795400

External links

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• `Topology and groupoids', by Ronald Brown
• `Categories and groupoids', by P.J. Higgins
• `Higher dimensional group theory' gives a broad view of higher dimensional van Kampen theorems involving multiple groupoids.

Algebraic topology | Homotopy theory | Mathematical theorems

Théorème de van Kampen | Teorema di Van Kampen