(Continued from https://mathtuition88.com/2015/06/18/the-groupoid-properties-of-on-path-homotopy-classes/)

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.

Proof:

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)

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