By Brayton Gray
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Additional info for Homotopy theory: An Introduction to Algebraic Topology (Pure and Applied Mathematics 64)
Choose x , y E X and consider all paths p : Z -+ X with p ( 0 ) = x and p(1) = y (Fig. 1). Write n ( X ; x , y ) for the set of all homotopy classes of such paths relative to the end points. ) Thus n ( X ; x, x ) = nl(X, x ) . Our understanding of nl(X, *) is greatly facilitated by an organization of its elements into a group, which we now describe. We will define the composition of two based paths. This will induce a composition among the path classes. More generally, suppose we are given two paths p1 and p z subject only to the requirement thatpz(0) =pl(l).
X, and relations rl(xl,. , x,,) = 1, . . , rk(xl, . . , x,) = 1, and G2 is defined by generators y,, . . , y m and relations sl(yl, . . , y,) = 1, . . , s,(y,, . . , y,) = 1. Suppose finally that G is generated by zl, . . , zj. Then G, *G G, has as generators XI, and as relations r,, .. , X n > Y 1 , . . 3 Ym 7 . . , r k , s,, . . , s l , andfi(zi)f2 (zi)-' for 1 I i 5 j . 7. Calculating the Fundamental Group 44 Proof Let G be the group defined by these generators and relations above.
1 as follows. ) a, 7 Calculating the Fundamental Group We have done nothing, so far, to calculate nl(X) except in the most trivial cases. In this section we shall consider two methods of calculating n, and give some applications. The first method (covering spaces) is quite geometric and allows one to work from a conjecture based on intuition to the answer. It is absolutely useless in proving that a space is simply connected. The second method (the Van Kampen theorem) is analytical and somewhat more complicated, but can be easily used to show that spaces (such as S" for n 2 1) are simply connected.