0.2. The Seifert-van Kampen Theorem 5
bouquet of circles (Figure 0.1). Decompose X into open sets U1,U2,...,Un,
each homotopy equivalent to a circle, any two of which intersect in a fixed
contractible neighborhood of x0, and apply the theorem inductively to see
that π1(X, x0) is a free group with one generator for each of the 1-cells in X.
To be specific, let αi be the loop that goes once around ai. Then π1(X, x0)
is the free group generated by {α1,α2,...,αn}.
Now attach 2-cells b1,b2,...,bk to X to form a 2-dimensional CW com-
plex Y . Each 2-cell bi is attached via a map fi :
∂I2
→ X. We can think of fi
as representing an element of π1(X, x0), and [fi] can be written as a word βi
in α1,α2,...,αn. Applying the classical Seifert-van Kampen Theorem in-
ductively yields that π1(Y, x0) is isomorphic to the group with presentation
α1,α2,...,αn : β1,β2,...,βk .
Often we will encounter subsets of manifolds that do not have the ho-
motopy type of finite CW complexes. Such sets, in general, can have more
complicated fundamental groups than those computable via the Seifert-
vanKampen Theorem.
Example 0.2.5. Let cn be the circle of radius 1/n centered at the point
1/n, 0 in
R2.
Each cn passes through the base point z0 = 0, 0 . The
Hawaiian earring is the compact set Z = ∪n=1cn
∞
(see Figure 0.2). For
each n there is a loop γn that wraps once around cn. Even though Z looks
superficially like a straightforward generalization of the X in the previous
example, the group π1(Z, z0) is not generated by {γn}. To see this, note
that the loop β : [0, 1] → Z that wraps the subinterval [1/(n + 1), 1/n]
once around cn defines an element of π1(Z, x0) that cannot be written as a
finite product of γn’s. The structure of π1(Z, z0) is surprisingly large and
complex (and interesting); a detailed description of the group can be found
in (Cannon and Conner, 2000).
Figure 0.2. The Hawaiian earring
