Seifert–van Kampen theorem

From Wikipedia, the free encyclopedia

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, $π 1 (U, w)$ , $π 1 (V, w)$ , and $π 1 (W, w)$ , together with the inclusion homomorphisms (induced by the inclusion map):

$I\colon \pi_1(W,w)\to \pi_1(U,w)$

and

$J\colon \pi_1(W,w)\to \pi_1(V,w)$

are sufficient data to determine $π 1 (X, w)$ . The maps $I$ and $J$ extend to an epimorphism

$\Phi\colon\pi_1(U,w)\ast\pi_1(V,w)\to\pi_1(X,w)$

where $\pi_1(U,w)\ast\pi_1(V,w)$ is the free product of $π 1 (U, w)$ and $π 1 (V, w)$ . 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 $π 1 (X, w)$ is therefore isomorphic to $\pi_1(U,w)\ast\pi_1(V,w)$ modulo such elements, more precisely, to the almagamated free product: $\scriptstyle \pi_1(U,w)\ast_{\pi(W,w)}\pi_1(V,w)$

In particular, when $W$ is simply connected (so that its fundamental group is the trivial group), the theorem says that $π 1 (X, w)$ is isomorphic to the free product $\pi_1(U,w)\ast\pi_1(V,w)$ .

[edit] 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.

[edit] 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 the interiors of two path connected subspaces $X 1$ , $X 2$ . Suppose $X_0:= X_1\cap X_2$ is path connected. Let also $* \in ~ X_0$ and $i_k:\pi_1(X_0,*) \to \pi_1(X_k,*)$ , $j_k:\pi_1(X_k,*) \to \pi_1(X,*)$ be induced by the inclusions for k= 1, 2. Then X is path connected and the natural morphism $\pi_1(X_1,*)*{\pi_1(X_0,*)}\pi_1(X_2,*)\to \pi_1(X,*)$ is an isomorphism, that is, the fundamental group of X is the free product of the fundamental groups of $X 1$ and $X 2$ with amalgamation of $π 1 (X 0, * )$ .

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 $π 1 (X, A)$ 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 $π 1 (X, A)$ 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].

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]

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 $U λ$ , 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.

[edit] 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 $π 1 (S 2) = π 1 (A) * π 1 (B) / k e r (Φ)$ . 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 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 $π 1 (A) = π 1 (D 2) = 1$ . Thus the inclusion of $\pi_1(A \cap B)$ into $π 1 (A)$ sends any generator to the trivial element. However, the inclusion of $\pi_1(A \cap B)$ into $π 1 (B)$ is not trivial. In order to understand this, first one must calculate $π 1 (B)$ . 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

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

[edit] 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. 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 three fold 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.

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 $Π(X)$ , 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 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].

[edit] See also

[edit] 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. 2008. Higher dimensional, higher homotopy, generalized van Kampen Theorem., in Nonabelian Algebraic Topology: Higher homotopy groupoids of filtered spaces., Part III. 459 pp, (Preprint).
^ 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

[edit] 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, Philip J. Higgins and Rafael Sivera. 2008. Higher dimensional, higher homotopy, generalized van Kampen Theorem., in Nonabelian Algebraic Topology: Higher homotopy groupoids of filtered spaces. Part III. 516 pp, (Preprint).
Ronald Brown, Philip J. Higgins and Rafael Sivera. 2009. Nonabelian Algebraic Topology: Higher homotopy groupoids of filtered spaces., 516 pp, Preprint. http://planetphysics.org/?op=getobj&from=books&id=249
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).
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 Non-Abelian Algebraic Topology, (vol. 2 in preparation); downloadable PDF:
Van Kampen's theorem on PlanetMath
Van Kampen's theorem result on PlanetMath
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, Simplicial Sets and van Kampen's Theorem (Discusses generalized versions of van Kampen's theorem applied to topological spaces and simplicial sets).

[0] 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

[1] ttp://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.

[2] Ronald Brown. "Higher dimensional group theory". 2007. http://www.bangor.ac.uk/~mas010/hdaweb2.htm

[3] R. Brown. Topology and Groupoids., Booksurge PLC (2006). http://www.bangor.ac.uk/~mas010/topgpds.html

[4] ttp://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.

[5] R. Brown, Topology and Groupoids., Booksurge PLC (2006).

[6] Ronald Brown, Philip J. Higgins and Rafael Sivera. 2008. Higher dimensional, higher homotopy, generalized van Kampen Theorem., in Nonabelian Algebraic Topology: Higher homotopy groupoids of filtered spaces., Part III. 459 pp, (Preprint).

[7] Higher dimensional, generalized van Kampen theorems (HD-GVKT) http://planetphysics.org/encyclopedia/HDGvKTVanKampenTheorems.html

[8] Ronald Brown. "Higher dimensional group theory" . 2007. http://www.bangor.ac.uk/~mas010/hdaweb2.htm

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Seifert–van Kampen theorem

From Wikipedia, the free encyclopedia

Contents

[edit] Equivalent formulations

[edit] Van Kampen's theorem for fundamental groups

[edit] Examples

[edit] Generalizations

[edit] See also

[edit] Notes

[edit] References

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