Theorem: The operation * has the following properties:
(1) (Associativity) [f]*([g]*[h])=([f]*[g])*[h], i.e. it doesn’t matter where we place the brackets.
(2) (Right and left identities) Given , let denote the constant path mpping all of I to the point x. If f is a path in X from to , then and .
(3) (Inverse) Given the path f in X from to , let be the path defined by . is called the reverse of f. Then, and .
We will prove the above statements, of which (1) Associativity is actually the trickiest.
We shall prove two elementary lemmas first. (This part is not proved in the book by Munkres).
Lemma 1: If is a continuous map, and if F is a path homotopy in X between the paths f and f’, then is a path homotopy in Y between the paths and .
Proof of Lemma 1: Since F is a path homotopy in X between paths f and f’, we have by definition that F(s,0)=f(s), F(s,1)=f'(s), F(0,t)=x_0, F(1,t)=x_1.
Then, k F(s,0)=kf(s), kF(s,1)=kf'(s), kF(0,t)=k(x_0), kF(1,t)=k(x_1). Since kF is continuous (composition of two continuous functions), kF is inded a path homotopy in Y between he paths kf and kf’.
Lemma 2: If is a continuous map and if f and g are paths in X with f(1)=g(0), then
Proof of Lemma 2:
, where h=f*g as defined previously.
We will first verify property (2) on Right and Left Identities. Let denote the constant path in I at 0, and we let denote the identity map, which is a path in I from 0 to 1. Then is also a path in I from 0 to 1.
Because I is convex, there is a path homotopy G in I between i and (Straight-line homotopy) Then is a path homotopy in X between the paths and (Lemma 1). Furthermore by Lemma 2, which is equivalent to .
A similar argument, using the fact that if denotes the constant path at 1, then is path homotopic in I to the path i, shows that .
To prove (3) (Inverse), we note that the reverse of i is . Then is a path in I beginning and ending at 0. The constant path is also beginning and ending at 0. Again, because I is convex, there is a path homotopy H in I between and (straight-line homotopy). Then, using lemma 1 and 2, is path homotopy between and . Very similarly, we can use the fact that is path homotopic in I to to show that .
We will continue the proof of associativity (which is longer) in the next blog post.
Source: Topology (2nd Economy Edition)