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) = <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 | 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.

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.

Examples

One can use Van Kampen's theorem to calculate fundamental groups for topological spaces that can be decomposed into simpler spaces. For example, consider the sphere $S^2$. Pick open sets $A=S^2-\{n\}$ and $B=S^2-\{s\}$ where n and s denote the north and south poles respectively. Then we have the property that A, B and A ⋂ B are open path connected sets. Thus we can see that there is a commutative diagram including A ⋂ B into A and B and then another inclusion from A and B into $S^2$ and that there is a corresponding diagram of homomorphisms between the fundamental groups of each subspace. Applying Van Kampen's theorem gives the result $pi\_1(S^2)=pi\_1(A)*pi\_1(B)/ker(\{Phi\})$. However A and B are both homeomorphic to $mathbf\{R^2\}$ which is simply connected, so both A and B have trivial fundamental groups. It is clear from this that the fundamental group of $S^2$ is trivial.

A more complicated example is the calculation of the fundamental group of a genus "n" orientable surface "S", otherwise known as the "genus n surface group". One can construct "S" using its standard fundamental polygon. For the first open set "A", pick a disk within the center of the polygon. Pick "B" to be the compliment in "S" of the center point of "A". Then the intersection of "A" and "B" is an annulus, which is known to be homotopy equivalent to (and so has the same fundamental group as) a circle. Then $pi\_1(A\; cap\; B)=pi\_1(S^1)$, which is the integers, and $pi\_1(A)=pi\_1(D^2)=\{1\}$. Thus the inclusion of $pi\_1(A\; cap\; B)$ into $pi\_1(A)$ sends any generator to the trivial element. However, the inclusion of $pi\_1(A\; cap\; B)$ into $pi\_1(B)$ is not trivial. In order to understand this, first one must calculate $pi\_1(B)$. This is easily done as one can deformation retract B (which is "S" with one point deleted) onto the edges labeled by "A₁B₁A₁^-1B₁^-1A₂B₂A₂^-1B₂^-1... A_nB_nA_n^-1B_n^-1". This space is known to be the wedge sum of 2"n" circles (also called a bouquet of circles), which further is known to have fundamental group isomorphic to the free group with 2"n" generators, which in this case can be represented by the edges themselves: $\{A\_1,B\_1,ldots,A\_n,B\_n\}$. We now have enough information to apply Van Kampen's theorem. The generators are the loops $\{A\_1,B\_1,ldots,A\_n,B\_n\}$ ("A" is simply connected, so it contributes no generators) and there is exactly one relation: "A₁B₁A₁^-1B₁^-1A₂B₂A₂^-1B₂^-1... A_nB_nA_n^-1B_n^-1" = 1. Using generators and relations, this group is denoted

:$langle\; A\_1,B\_1,ldots,A\_n,B\_n|A\_1B\_1A\_1^\{-1\}B\_1^\{-1\}ldots\; A\_nB\_nA\_n^\{-1\}B\_n^\{-1\}\; angle.$

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 further by considering the fundamental groupoid $Pi(X)$, 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.

Applications in Physics

Potential applications of the higher homotopy, generalised van Kampen's theorem are in Theoretical Physics, especially in quantum Field Theories,Supergravity, Supersymmetry, and classification of Quantum Space-time invariants [Brown et al. 2007] .

References

* Allen Hatcher, [http://www.math.cornell.edu/~hatcher/AT/ATpage.html "Algebraic topology."] (2002) Cambridge University Press, Cambridge, xii+544 pp. ISBN 052179160X and ISBN 0521795400
* Peter May, [http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf "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, " [http://www.bangor.ac.uk/r.brown/topgpds.html Topology and groupoids] " (2006) Booksurge LLC ISBN 1-4196-2722-8
*P.J. Higgins, " [http://138.73.27.39/tac/reprints/articles/7/tr7abs.html Categories and groupoids] " (1971) Van Nostrand Reinhold
*Ronald Brown, " [http://www.bangor.ac.uk/r.brown/hdaweb2.html Higher dimensional group theory] " (2007) "(Gives a broad view of higher dimensional van Kampen theorems involving multiple groupoids.)"
*E. R. van Kampen. "On the connection between the fundamental groups of some related spaces." American Journal of Mathematics, vol. 55 (1933), pp. 261–267.
*Ronald Brown, Higgins, P. J. and R. Sivera. 2007, vol. 1 [http://www.bangor.ac.uk/~mas010/nonab-a-t.html "Non-Abelian Algebraic Topology"] , (vol. 2 "in preparation"); [http://www.bangor.ac.uk/~mas010/nonab-t/partI010604.pdf downloadable PDF:]
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*Ronald Brown R, K. Hardie, H. Kamps, T. Porter T.: The homotopy double groupoid of a Hausdorff space., "Theory Appl. Categories", 10:71–-93 (2002).
* Dylan G.L. Allegretti, [http://www.math.uchicago.edu/~may/VIGRE/VIGREREU2008.html "Simplicial Sets and van Kampen's Theorem."] 1. Ronald Brown, J. F. Glazebrook and I. C. Baianu: A categorical and higher dimensional algebra framework for complex systems and spacetime structures, "Axiomathes", 17 :409--493 (2007).