Seifert–Van Kampen theorem

in terms of the fundamental groups of two open, path-connected subspaces that coverIt can therefore be used for computations of the fundamental group of spaces that are constructed out of simpler ones.Let X be a topological space which is the union of two open and path connected subspaces U1, U2.Suppose U1 ∩ U2 is path connected and nonempty, and let x0 be a point in U1 ∩ U2 that will be used as the base of all fundamental groups.form a commutative pushout diagram: The natural morphism k is an isomorphism.[1] 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.Unfortunately, the theorem as given above does not compute the fundamental group of the circle – which is the most important basic example in algebraic topology – because the circle cannot be realised as the union of two open sets with connected intersection.on a set A of base points, chosen according to the geometry of the situation.This theorem gives the transition from topology to algebra, in determining completely the fundamental groupoidThe interpretation of this theorem as a calculational tool for "fundamental groups" needs some development of 'combinatorial groupoid theory'.There is a version of the last theorem when X is covered by the union of the interiors of a family[7][8] 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.[...] people still obstinately persist, when calculating with fundamental groups, in fixing a single base point, instead of cleverly choosing a whole packet of points which is invariant under the symmetries of the situation, which thus get lost on the way.In certain situations (such as descent theorems for fundamental groups à la Van Kampen) it is much more elegant, even indispensable for understanding something, to work with fundamental groupoids with respect to a suitable packet of base points [...]In the language of combinatorial group theory, ifis the pushout, in the category of groups, of the diagram: One can use Van Kampen's theorem to calculate fundamental groups for topological spaces that can be decomposed into simpler spaces.Applying Van Kampen's theorem gives the result However, A and B are both homeomorphic to R2 which is simply connected, so both A and B have trivial fundamental groups.are CW complexes), then we can apply the Van Kampen theorem toFor the first open set A, pick a disk within the center of the polygon.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.This is easily done as one can deformation retract B (which is S with one point deleted) onto the edges labeled by 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:[10] As explained above, this theorem was extended by Ronald Brown to the non-connected case by using the fundamental groupoid[11] The theorem and proof for the fundamental group, but using some groupoid methods, are also given in J. Peter May's book.[12] 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.Applications of the fundamental groupoid on a set of base points to the Jordan curve theorem, covering spaces, and orbit spaces are given in Ronald Brown's book.[13] In the case of orbit spaces, it is convenient to take A to include all the fixed points of the action.[14] Thus a 2-dimensional Van Kampen theorem which computes nonabelian second relative homotopy groups was given by Ronald Brown and Philip J.[15] A full account and extensions to all dimensions are given by Brown, Higgins, and Rafael Sivera,[16] while an extension to n-cubes of spaces is given by Ronald Brown and Jean-Louis Loday.[17] 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 by descent theory.