If each intersection is path-connected for all , then the homomorphism induced by the inclusion maps is surjective.
If, additionally, each triple intersection is path-connected for all , then the kernel of is the normal subgroup generated by all elements of the form for .
In this case, induces an isomorphism:
The path-connectedness of the intersections is a crucial hypothesis. If it fails, the theorem does not apply and can give incorrect results:
e.g. Cover the circle with two open arcs and , where .
We have and since and are contractible.
A naive application of the theorem might suggest , which is false.
The issue is that the intersection consists of two disjoint arcs and is therefore not path-connected. Condition (1) of the theorem is not met.
e.g. Let be a sphere with two points identified. Let be the identification point. Consider an open cover by three sets , , and where are points on the two “lobes” and the “equator” respectively.
Each of deformation retracts to a circle, so .
The pairwise intersections , , and are contractible, thus path-connected.
Van Kampen part (1) would suggest is a quotient of .
However, the space is the wedge sum , so its fundamental group is .
The issue here is that the triple intersection is not path-connected, so condition (2) does not apply.
For the theorem to extend to infinite unions of closed sets, a local condition on the basepoint is often required.
Good Point
A point is called good if has an open neighborhood that deformation retracts to .
e.g. Consider the space , where is the circle in with radius centered at . The wedge point where all circles meet is not a “good” point. Any open neighborhood of contains infinitely many of the circles and does not deformation retract to .
Fundamental Group of a Wedge Sum
Suppose is a good point for each . Then the fundamental group of the wedge sum is the free product of the fundamental groups of the components:
Proof Sketch One can “fatten” each space to an open set in the wedge sum such that deformation retracts to . The pairwise and triple intersections of these fattened sets will deformation retract to the wedge point . Since is a good point, these intersections are path-connected. Van Kampen’s theorem then applies to give the result.
Applications: Complements of Links in
Complement of a Single Circle
Let . This space deformation retracts to a 2-sphere union one of its diameter that pass through the removed . There are two approaches to think about this. One way is that we can explicitly construct a retraction as follows: for any , if is outside of the sphere, we map it to ; otherwise, is inside the sphere, we pick its closest point on the removed , and travel away from until we reach either the diameter or the sphere.
Another way is to note that , which is the complement of a solid torus. One can visualize this by embedding the solid torus inside a 2-sphere, and then inflating the torus until it nearly touches the sphere’s surface. The space outside the torus then deformation retracts onto a 2-sphere together with one of its diameters passing through the “hole” of the torus.
Now let denote the deformation retract obtained from either approach. We can cover with two open sets: Note that is homeomorphic to , and is the green arc, which is contractible, so The generator is a loop that links with the removed circle .
Complement of Unlinked Circles
Let , where and are two unlinked circles. By applying the Seifert-van Kampen theorem, one can show that the fundamental group is the free product of two copies of .
This group is non-abelian. A loop winding around the first circle does not commute with a loop winding around the second. The space has the homotopy type of .
Complement of Linked Circles (Hopf Link)
Let , where and form a simple link (the Hopf link).
The fundamental group is the direct product (or sum) of two copies of :
This group is abelian. The linking of the circles forces the generators of the fundamental group to commute. This space is claimed to be homotopy equivalent to , which has the correct fundamental group.
Now let