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 also assume that $U$ and $V$ are open subspaces with union $X$ .

1 Equivalent formulations
2 Van Kampen's theorem for fundamental groups
3 Examples
4 Generalizations
5 See also
6 Notes
7 References

Equivalent formulations

In the language of combinatorial group theory, $\pi_1(X,w)$ is the free product with amalgamation of $\pi_1(U,w)$ and $\pi_1(V,w)$ , with respect to the (not necessarily injective) homomorphisms $I$ and $J$ . Given group presentations:

$\pi_1(U,w) = \langle u_1,...,u_k | \alpha_1,...,\alpha_l\rangle$

$\pi_1(V,w) = \langle v_1,...,v_m | \beta_1,...,\beta_n\rangle$ , and

$\pi_1(W,w) = \langle w_1,...,w_p | \gamma_1,...,\gamma_q\rangle$

the amalgamation can be presented as

$\pi_1(X,w) = \langle u_1,...,u_k, v_1,...,v_m | \alpha_1,...,\alpha_l, \beta_1,...,\beta_n, I(w_1)J(w_1)^{-1},...,I(w_p)J(w_p)^{-1}\rangle$ .

In category theory, $\pi_1(X,w)$ is the pushout, in the category of groups, of the diagram:

$\pi_1(U,w)\gets\pi_1(W,w)\to\pi_1(V,w)$ .

Van Kampen's theorem for fundamental groups

Van Kampen's theorem for fundamental groups^[1]:

Let X be a topological space which is the union of two open and path connected subspaces $U_1$ , $U_2$ . Suppose $U_1\cap U_2$ is path connected and let x₀ be a point in it that will be used as the base of all fundamental groups, then X is path connected and the inclusion morphisms draw a commutative pushout diagram:

the natural morphism k is an isomorphism, that is, the fundamental group of X is the free product of the fundamental groups of $U_1$ and $U_2$ with amalgamation of $\pi_1(U_1\cap U_2, x_0)$ .

Usually the morphisms induced by inclusion in this theorem are not themselves injective, and the more precise version of the statement is in terms of pushouts of groups. The notion of pushout in the category of groupoids allows for a version of the theorem for the non path connected case, using the fundamental groupoid $\pi_1(X,A)$ on a set A of base points,.^[2] This groupoid consists of homotopy classes relative to the end points of paths in X joining points of $A\cap X$ . In particular, if X is a contractible space, and A consists of two distinct points of X, then $\pi_1(X,A)$ is easily seen to be isomorphic to the groupoid often written $\mathcal I$ with two vertices and exactly one morphism between any two vertices. This groupoid plays a role in the theory of groupoids analogous to that of the group of integers in the theory of groups.^[3] The groupoid $\mathcal I$ also allows for groupoids a notion of homotopy: it is a unitinterval object in the category of groupoids.

Theorem: Let the topological space X be covered by the interiors of two subspaces $X_1, X_2$ and let A be a set which meets each path component of $X_1, X_2$ and $X_0:=X_1 \cap X_2$ , then A meets each path component of X and the diagram P of morphisms induced by inclusion

is a pushout diagram in the category of groupoids.^[4]

To see its utility, one can easily find cases where X is connected but is the union of the interiors of two subspaces, each with say 402 path components and whose intersection has say 1004 path components. The interpretation of this theorem as a calculational tool for fundamental groups needs some development of `combinatorial groupoid theory',.^[5]^[6] This theorem implies the calculation of the fundamental group of the circle as the group of integers, since the group of integers is obtained from the groupoid $\mathcal I$ by identifying, in the category of groupoids, its two vertices.

There is a version of the last theorem when X is covered by the union of the interiors of a family $\{U_\lambda�: \lambda \in \Lambda\}$ of subsets.^[7]^[8] The conclusion is that if A meets each path component of all 1,2,3-fold intersections of the sets $U_\lambda$ , then A meets all path components of X and the diagram $\bigsqcup_{(\lambda,\mu) \in \Lambda^2} \pi_1(U_\lambda \cap U_\mu, A) \rightrightarrows \bigsqcup_{\lambda \in \Lambda} \pi_1(U_\lambda, A)\rightarrow \pi_1(X,A)$ of morphisms induced by inclusions is a coequaliser in the category of groupoids.

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 $A \cap B$ are open path connected sets. Thus we can see that there is a commutative diagram including $A \cap B$ 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 $\mathbb{R}^2$ 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 complement 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 2n circles (also called a bouquet of circles), which further is known to have fundamental group isomorphic to the free group with 2n 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

$\langle A_1,B_1,\ldots,A_n,B_n|A_1B_1A_1^{-1}B_1^{-1}\ldots A_nB_nA_n^{-1}B_n^{-1}\rangle.$

Generalizations

This theorem has been extended to the non-connected case by using the fundamental groupoid $\pi_1 (X,A)$ 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. For example, if the intersection W has two path components, it is convenient to let A consist of one point in each of these components. A theorem for arbitrary covers, with the restriction that A meets all threefold intersections of the sets of the cover, is given in the paper by Brown and Razak cited below. Applications of the fundamental groupoid on a set of base points to the Jordan curve theorem, Covering space, and orbit space are given in Ronald Brown's book cited below. See the book Topology and groupoids listed below.

In the case of orbit spaces, it is convenient to take A to include all the fixed points of the action. An example here is the conjugation action on the circle.

The version that allows more than two overlapping sets but with A a singleton is also given in Allen Hatcher's book below, theorem 1.20.

In fact, we can extend van Kampen's theorem significantly further by considering the fundamental groupoid $\pi(X)$ , a small category whose objects are points of X and whose arrows are homotopy equivalences of paths. In this case, to determine the fundamental groupoid of a space, we need only know the fundamental groupoids of a covering of the space by path-connected components: create a new category in which the objects are fundamental groupoids of the path-connected open sets that form the cover, 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. One can think of the colimit relation as a disjoint union followed by a quotient. A precise statement of the theorem along with proof is given in Peter May's book A Concise Introduction to Algebraic Topology, chapter 2.

References to higher dimensional versions of the theorem which yield some information on homotopy types are given in an article on higher dimensional group theories and groupoids.^[9]

Fundamental groups also appear in algebraic geometry and are the main topic of Alexander Grothendieck's first Séminaire de géométrie algébrique (SGA1). A version of van Kampen's theorem appears there, and is proved along quite different lines than in algebraic topology, namely descent theory. A similar proof works in algebraic topology, see ^[10].

Notes

^ R. Brown, Groupoids and Van Kampen's theorem, Proc. London Math. Soc. (3) 17 (1967) 385-401. http://planetmath.org/?method=src&from=objects&name=VanKampensTheorem&op=getobj
^ http://planetmath.org/?method=src&from=objects&name=VanKampensTheorem&op=getobj R. Brown, Groupoids and Van Kampen's theorem, Proc. London Math. Soc. (3) 17 (1967) 385-401.
^ Ronald Brown. "Higher dimensional group theory". 2007. http://www.bangor.ac.uk/~mas010/hdaweb2.htm
^ R. Brown. Topology and Groupoids., Booksurge PLC (2006). http://www.bangor.ac.uk/~mas010/topgpds.html
^ http://planetmath.org/?method=src&from=objects&name=VanKampensTheorem&op=getobj P.J. Higgins, Categories and Groupoids, van Nostrand, 1971, Reprints of Theory and Applications of Categories, No. 7 (2005),pp 1-195.
^ R. Brown, Topology and Groupoids., Booksurge PLC (2006).
^ Ronald Brown, Philip J. Higgins and Rafael Sivera. Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy groyupoids, European Mathematical Society Tracts vol 15, August, 2011.
^ Higher dimensional, generalized van Kampen theorems (HD-GVKT) http://planetphysics.org/encyclopedia/HDGvKTVanKampenTheorems.html
^ Ronald Brown. "Higher dimensional group theory" . 2007. http://www.bangor.ac.uk/~mas010/hdaweb2.htm
^ A. & R. Douady. "Algèbre et théories galoisiennes". Cassini (2005)

This article incorporates material from Van Kampen's theorem, which is licensed under the Creative Commons Attribution/Share-Alike License.

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).
Higher dimensional algebra
Ronald Brown, Topology and groupoids (2006) Booksurge LLC ISBN 1-4196-2722-8
R. Brown and A. Razak, ``A van Kampen theorem for unions of non-connected spaces, Archiv. Math. 42 (1984) 85-88.
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).
Seifert, H., Konstruction drei dimensionaler geschlossener Raume. Berichte Sachs. Akad. Leipzig, Math.-Phys. Kl. (83) (1931) 26-66.
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.
Brown, R., Higgins, P. J, On the connection between the second relative homotopy groups of some related spaces, Proc. London Math. Soc. (3) 36 (1978) 193-212.
Brown, R., Higgins, P. J. and Sivera, R.. 2011, EMS Tracts in Mathematics Vol.15 (2011) Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy groupoids; (The first of three Parts discusses the applications of the 1- and 2-dimensional versions of the Seifert-van Kampen Theorem. The latter allows calculations of nonabelian second relative homotopy groups, and in fact of homotopy 2-types. The second part applies a Higher Homotopy van Kampen Theorem for crossed complexes, proved in Part III.)
Van Kampen's theorem result on PlanetMath
R. Brown, H. Kamps, T. Porter : A homotopy double groupoid of a Hausdorff space II: a van Kampen theorem', Theory and Applications of Categories, 14 (2005) 200-220.
Dylan G.L. Allegretti, Simplicial Sets and van Kampen's Theorem (Discusses generalized versions of van Kampen's theorem applied to topological spaces and simplicial sets).

R. Brown and J.-L. Loday, ``Van Kampen theorems for diagrams of spaces, Topology 26 (1987) 311-334.