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=1 cn (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
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