NoteNextra-origin/content/Math4202/Math4202_L10.md

Math4202 Topology II (Lecture 10)

Algebraic Topology

Path homotopy

Theorem for properties of product of paths

1. If f\simeq_p f_1, g\simeq_p g_1, then f*g\simeq_p f_1*g_1. (Product is well-defined)
2. ([f]*[g])*[h]=[f]*([g]*[h]). (Associativity)
3. Let e_{x_0} be the constant path from x_0 to x_0, e_{x_1} be the constant path from x_1 to x_1. Suppose f is a path from x_0 to x_1.

   
   [e_{x_0}]*[f]=[f],\quad [f]*[e_{x_1}]=[f]
   

   (Right and left identity)
4. Given f in X a path from x_0 to x_1, we define \bar{f} to be the path from x_1 to x_0 where \bar{f}(t)=f(1-t).

   
   f*\bar{f}=e_{x_0},\quad \bar{f}*f=e_{x_1}
   

   
   [f]*[\bar{f}]=[e_{x_0}],\quad [\bar{f}]*[f]=[e_{x_1}]
   

Proof

(1) If f\simeq_p f_1, g\simeq_p g_1, then f*g\simeq_p f_1*g_1.

Let F be homotopy between f and f_1, G be homotopy between g and g_1.

We can define

F*G:[0,1]\times [0,1]\to X,\quad F*G(s,t)=\left(F(-,t)*G(-,t)\right)(s)=\begin{cases}
F(2s,t) & 0\leq s\leq \frac{1}{2}\\
G(2s-1,t) & \frac{1}{2}\leq s\leq 1
\end{cases}

F*G is a homotopy between f*g and f_1*g_1.

We can check this by enumerating the cases from definition of homotopy.

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(2) ([f]*[g])*[h]=[f]*([g]*[h]).

For f*(g*h), along the interval [0,\frac{1}{2}] we map x_1\to x_2, then along the interval [\frac{1}{2},\frac{3}{4}] we map x_2\to x_3, then along the interval [\frac{3}{4},1] we map x_3\to x_4.

For (f*g)*h, along the interval [0,\frac{1}{4}] we map x_1\to x_2, then along the interval [\frac{1}{4},\frac{1}{2}] we map x_2\to x_3, then along the interval [\frac{1}{2},1] we map x_3\to x_4.

We can construct the homotopy between f*(g*h) and (f*g)*h as follows.

Let f((4-2t)s) for F(s,t),

when t=0, F(s,0)=f(4s)\in f*(g*h), when t=1, F(s,1)=f(2s)\in (f*g)*h.

....

We make the linear maps between f*(g*h) and (f*g)*h continuous, then f*(g*h)\simeq_p (f*g)*h. With our homotopy constructed above

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(3) e_{x_0}*f\simeq_p f\simeq_p f*e_{x_1}.

We can construct the homotopy between e_{x_0}*f and f as follows.

H(s,t)=\begin{cases}
x_0 & t\geq 2s\\
f(2s-t) & t\leq 2s
\end{cases}

or you may induct from f(\frac{s-t/2}{1-t/2}) if you like.

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(4) f*\bar{f}=e_{x_0},\quad \bar{f}*f=e_{x_1}.

Note that we don't need to reach x_1 every time.

f_t=f(ts) s\in[0,\frac{1}{2}].

\bar{f}_t=\bar{f}(1-ts) s\in[\frac{1}{2},1].

Caution

Homeomorphism does not implies homotopy automatically.

Definition for the fundamental group

The fundamental group of X at x is defined to be

(\Pi_1(X,x),*)