The Groupoid Properties of Operation * on Path-homotopy Classes (Proof)

(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 x\in X, let e_x denote the constant path e_x: I\to X mpping all of I to the point x. If f is a path in X from x_0 to x_1, then [f]*[e_{x_1}]=[f] and [e_{x_0}]*[f]=[f].

(3) (Inverse) Given the path f in X from x_0 to x_1, let \bar{f} be the path defined by \bar{f}=f(1-s). \bar{f} is called the reverse of f. Then, [f]*[\bar{f}]=[e_{x_0}] and [\bar{f}]*[f]=[e_{x_1}].

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 k: X\to Y is a continuous map, and if F is a path homotopy in X between the paths f and f’, then k\circ F is a path homotopy in Y between the paths k\circ f and k\circ f'.

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 k:X\to Y is a continuous map and if f and g are paths in X with f(1)=g(0), then

k\circ (f*g)=(k\circ f)*(k \circ g)

Proof of Lemma 2:

k(f*g)(s)=kh(s), where h=f*g as defined previously.

(kf)*(kg)(s)=kh(s).

We will first verify property (2) on Right and Left Identities. Let e_{x_0} denote the constant path in I at 0, and we let i: I\to I denote the identity map, which is a path in I from 0 to 1. Then e_0 * i 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 e_0 *i (Straight-line homotopy) Then f\circ G is a path homotopy in X between the paths f\circ i=f and f\circ (e_0 *i) (Lemma 1). Furthermore by Lemma 2, f\circ (e_0 *i) = (f \circ e_0) * (f \circ i) which is equivalent to e_{x_0} *f.

A similar argument, using the fact that if e_1 denotes the constant path at 1, then i*e_i is path homotopic in I to the path i, shows that [f]*[e_{x_1}]=[f].

To prove (3) (Inverse), we note that the reverse of i is \bar{i}(s)=1-s. Then i*\bar{i} is a path in I beginning and ending at 0. The constant path e_0 is also beginning and ending at 0. Again, because I is convex, there is a path homotopy H in I between e_0 and i*\bar{i} (straight-line homotopy). Then, using lemma 1 and 2, f\circ H is path homotopy between f\circ e_0=e_{x_0} and f\circ (i*\bar{i})=(f\circ i)*(f\circ\bar{i})=f*\bar{f}. Very similarly, we can use the fact that \bar{i}*i is path homotopic  in I to e_1 to show that [\bar{f}]*[f]=[e_{x_1}].

We will continue the proof of associativity (which is longer) in the next blog post.

Source: Topology (2nd Economy Edition)

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2 Responses to The Groupoid Properties of Operation * on Path-homotopy Classes (Proof)

  1. Pingback: Proof of Associativity of Operation * on Path-homotopy Classes | Singapore Maths Tuition

  2. Pingback: The Fundamental Group | Singapore Maths Tuition

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