Math4202 Topology II (Lecture 8)

Algebraic Topology

Let f,f':I\to X be a continuous maps from an interval I=[0,1] to a topological space X.

Two pathes f and f' are path homotopic if

there exists a continuous map F:I\times [0,1]\to X such that F(i,0)=f(i) and F(i,1)=f'(i) for all i\in I.
F(s,0)=f(0) and F(s,1)=f(1), \forall s\in I.$F(s,0)=f(0)$ and F(s,1)=f(1), \forall s\in I

The \simeq, \simeq_p are both equivalence relations.

Proof

Reflexive:

f:I\to X, F:I\times I\to X, F(s,t)=f(s).

F is a homotopy between f and f itself.

Symmetric:

Suppose f,g:I\to X,

F:I\times I\to X is a homotopy between f and g.

Let H: I\times I\to X be a homotopy between g and f defined as follows:

H(s,t)=F(s,1-t).

H(s,0)=F(s,1)=g(s), H(s,1)=F(s,0)=f(s).

Therefore H is a homotopy between g and f.

Transitive:

Suppose we have f\simeq_p g with homotopy F_1, and g\simeq_p h with homotopy F_2.

Then we can glue the two homotopies together to get a homotopy F between f and h using pasting lemma.

$F(s,t)=(F_1*F_2)(s,t)\coloneqq\begin{cases} F_1(s,2t), & t\in [0,\frac{1}{2}]\ F_2(s,2t-1), & t\in [\frac{1}{2},1] \end{cases}$

Therefore f\simeq_p h with homotopy F.

Note

We use [x] to denote the equivalence class of x.

Example of equivalence classes in path homotopy

Let X=\{pt\}, \operatorname{Path}(X)=\{\text{constant map}\}.$\operatorname{Path}/_{\simeq_p}(X)={[\text{constant map}]}$.

X=\{p,q\} with discrete topology, \operatorname{Path}(X)=\{f_{p},f_{q}\}.$\operatorname{Path}/{\simeq_p}(X)={[f{p}], [f_{q}]}$

This applied to all discrete topological spaces.

Let X=\mathbb{R} with standard topology.

\operatorname{Path}(X)=\{f:[0,1]\to \mathbb{R}\in C^0\}

Let f_1,f_2:[0,1]\to \mathbb{R} where f_1(0)=f_2(0), f_1(1)=f_2(1).

Then we can construct a homotopy between f_1 and f_2.

F:[0,1]\times [0,1]\to \mathbb{R}, F(s,t)=(1-t)f_1(s)+tf_2(s) is a homotopy between f_1 and f_2.

\operatorname{Path}/_{\simeq_p}(X)=\{(x_1,x_1)|x_1,x_2\in \mathbb{R}\}

This applies to any convex space V in \mathbb{R}^n.