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