Van Kampen's theorem and presentations

Suppose $X=U\cup V$, where U and V are open and path connected, their intersection is path connected, and $x_0\in U\cap V$. Let the two inclusions of the intersection induce homomorphisms into the groups of U and V. The fundamental group of X is their pushout: start with the free product and impose equality between the two images of each intersection loop. The inclusion homomorphisms need not be injective. Referring to an amalgamated free product must not hide an unsupported assumption that the overlap group embeds in both factors.

The proof subdivides loops and homotopies into pieces lying in U or V, using compactness of the parameter interval or square and the open cover. Paths inside the connected intersection transport subdivision endpoints back to the shared basepoint. These steps explain why the hypotheses appear. If the overlap is disconnected, this simple one-basepoint statement cannot be applied blindly; a groupoid version or additional paths can handle the missing information. Covering a circle by two contractible arcs with disconnected overlap, for instance, must not yield a falsely trivial fundamental group.

A connected graph can be reduced using a maximal tree, leaving one loop generator for each edge outside the tree. Its fundamental group is free on those generators. Attaching a two-cell along a based loop w makes that loop contractible through the new disk. Van Kampen shows that the new group is the old group modulo the normal closure of w, not merely modulo the set containing w. Changing the attaching loop's basepoint conjugates the word and therefore leaves its normal closure unchanged.

For a connected CW complex, the two-skeleton determines its fundamental group. Cells of dimension at least three do not change that group, though they can change higher homotopy information. These statements follow from suitable open neighborhoods of the skeleta and cellular arguments; the cells themselves are not automatically an open cover to which the theorem applies directly. A presentation records topology through chosen generators and attaching maps, and different choices can produce quite different-looking presentations of isomorphic groups. Solving a presentation still requires algebra rather than simply counting cells.

A wedge of two circles has free group on a and b because the two loops can be traversed in arbitrary reduced order. Attaching a disk along the commutator $aba^{-1}b^{-1}$ forces a and b to commute. This produces the torus presentation and an abelian group of rank two. In contrast, attaching a disk along $a^m$ creates a finite-order relation on a. The topological picture of one disk is insufficient to distinguish these cases; its attaching map determines what algebraic information is removed.

A nullhomotopic attaching map imposes the trivial relation and leaves the fundamental group unchanged, even though it adds a sphere-like higher-dimensional feature. Reversing the orientation of an attaching loop replaces a relation by its inverse, which generates the same normal subgroup. These checks are useful when a diagram's arrows or polygon boundary conventions differ between sources. Write the traversal order and basepoint paths explicitly before simplifying, and verify the group with a homomorphism to a familiar model whenever a relation appears to eliminate more than intended.