Seifert–van Kampen theorem
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
, in terms of the fundamental groups of two open, path-connected subspaces 
and 
that cover 
. 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
can be partitioned into journeys: through the intersection 
of 
and 
, through 
but outside 
, and through 
outside 
. In order to move segments of paths around, by homotopy
to form loops returning to a base point
in 
, we should assume 
, 
and 
are path-connected and that 
isn't empty. We also assume that 
and 
are open subspaces with union 
.
,
is the free product with amalgamation of 
and 
, with respect to the (not necessarily injective) homomorphisms 
and 
. Given group presentations:
, and
the amalgamation can be presented as
.
In category theory
,
is the pushout, in the category of groups, of the diagram:
.
Let X be a topological space which is the union of two open and path connected subspaces
,
. Suppose 
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
and 
with amalgamation of 
.
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
on a set A of base points,. This groupoid consists of homotopy classes relative to the end points of paths in X joining points of 
. In particular, if X is a contractible space, and A consists of two distinct points of X, then 
is easily seen to be isomorphic to the groupoid often written 
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. The groupoid 
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
and let A be a set which meets each path component of 
and 
. 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.
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',. 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
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
of subsets. The conclusion is that if A meets each path component of all 1,2,3-fold intersections of the sets 
, then A meets all path components of X and the diagram 
of morphisms induced by inclusions is a coequaliser in the category of groupoids.
. Pick open sets 
and 
where n and s denote the north and south poles respectively. Then we have the property that A, B and 
are open path connected sets. Thus we can see that there is a commutative diagram including 
into A and B and then another inclusion from A and B into 
and that there is a corresponding diagram of homomorphisms between the fundamental groups of each subspace. Applying Van Kampen's theorem gives the result 
. However A and B are both homeomorphic to 
which is simply connected, so both A and B have trivial fundamental groups. It is clear from this that the fundamental group of 
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
, which is the integers, and 
. Thus the inclusion of 
into 
sends any generator to the trivial element. However, the inclusion of 
into 
is not trivial. In order to understand this, first one must calculate 
. 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... AnBnAn−1Bn−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:
. We now have enough information to apply Van Kampen's theorem. The generators are the loops 
(A is simply connected, so it contributes no generators) and there is exactly one relation: A1B1A1−1B1−1A2B2A2−1B2−1... AnBnAn−1Bn−1 = 1. Using generators and relations, this group is denoted
, 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
, 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.
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 .
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Seifert
Herbert Seifert
Herbert Karl Johannes Seifert was a German mathematician known for his work in topology....
-van Kampen
Egbert van Kampen
Egbert Rudolf van Kampen was a mathematician. He made important contributions to topology, especially to the study of fundamental groups....
theorem of algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, sometimes just called van Kampen's theorem, expresses the structure of the fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
of a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
, in terms of the fundamental groups of two open, path-connected subspaces and that cover . 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
can be partitioned into journeys: through the intersection of and , through but outside , and through outside . In order to move segments of paths around, by homotopyHomotopy
In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...
to form loops returning to a base point
in , we should assume , and are path-connected and that isn't empty. We also assume that and are open subspaces with union .Equivalent formulations
In the language of combinatorial group theoryCombinatorial group theory
In mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generators and relations...
,
is the free product with amalgamation of and , with respect to the (not necessarily injective) homomorphisms and . Given group presentations:, andthe amalgamation can be presented as
.In category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
,
is the pushout, in the category of groups, of the diagram:.Van Kampen's theorem for fundamental groups
Van Kampen's theorem for fundamental groups:Let X be a topological space which is the union of two open and path connected subspaces
,. Suppose 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
and with amalgamation of .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
on a set A of base points,. This groupoid consists of homotopy classes relative to the end points of paths in X joining points of . In particular, if X is a contractible space, and A consists of two distinct points of X, then is easily seen to be isomorphic to the groupoid often written 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. The groupoid 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
and let A be a set which meets each path component of and . 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.
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',. 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
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
of subsets. The conclusion is that if A meets each path component of all 1,2,3-fold intersections of the sets , then A meets all path components of X and the diagram 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. Pick open sets and where n and s denote the north and south poles respectively. Then we have the property that A, B and are open path connected sets. Thus we can see that there is a commutative diagram including into A and B and then another inclusion from A and B into and that there is a corresponding diagram of homomorphisms between the fundamental groups of each subspace. Applying Van Kampen's theorem gives the result . However A and B are both homeomorphic to which is simply connected, so both A and B have trivial fundamental groups. It is clear from this that the fundamental group of 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
, which is the integers, and . Thus the inclusion of into sends any generator to the trivial element. However, the inclusion of into is not trivial. In order to understand this, first one must calculate . This is easily done as one can deformation retractDeformation retract
In topology, a branch of mathematics, a retraction , as the name suggests, "retracts" an entire space into a subspace. A deformation retraction is a map which captures the idea of continuously shrinking a space into a subspace.- Retract :...
B (which is S with one point deleted) onto the edges labeled by A1B1A1−1B1−1A2B2A2−1B2−1... AnBnAn−1Bn−1. This space is known to be the wedge sum
Wedge sum
In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if X and Y are pointed spaces the wedge sum of X and Y is the quotient of the disjoint union of X and Y by the identification x0 ∼ y0:X\vee Y = \;/ \sim,\,where ∼ is the...
of 2n circles (also called a bouquet of circles
Bouquet of circles
In mathematics, a rose is a topological space obtained by gluing together a collection of circles along a single point. The circles of the rose are called petals. Roses are important in algebraic topology, where they are closely related to free groups.- Definition :A rose is a wedge sum of circles...
), which further is known to have fundamental group isomorphic to the free group
Free group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
with 2n generators, which in this case can be represented by the edges themselves:
. We now have enough information to apply Van Kampen's theorem. The generators are the loops (A is simply connected, so it contributes no generators) and there is exactly one relation: A1B1A1−1B1−1A2B2A2−1B2−1... AnBnAn−1Bn−1 = 1. Using generators and relations, this group is denotedGeneralizations
This theorem has been extended to the non-connected case by using the fundamental groupoid 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 theoremJordan curve theorem
In topology, a Jordan curve is a non-self-intersecting continuous loop in the plane, and another name for a Jordan curve is a "simple closed curve"...
, 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
, 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.
Fundamental groups also appear in algebraic geometry and are the main topic of Alexander Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
'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 .
See also
- Higher dimensional algebra
- Higher category theoryHigher category theoryHigher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities.- Strict higher categories :...
- Egbert van KampenEgbert van KampenEgbert Rudolf van Kampen was a mathematician. He made important contributions to topology, especially to the study of fundamental groups....
- Ronald Brown (mathematician)Ronald Brown (mathematician)Ronald Brown is an English mathematician. Emeritus Professor in the School of Computer Science at Bangor University, he has authored many books and journal articles.-Education and career:...