# Math4202 Topology II (Lecture 8)

## Algebraic Topology

### Path homotopy

#### Recall definition of path homotopy

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 $F(s,1)=f(1)$, $\forall s\in I$

#### Lemma: Homotopy defines an equivalence relation

The $\simeq$, $\simeq_p$ are both equivalence relations.

<details>
<summary>Proof</summary>

**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 $F$.

</details>

> [!NOTE]
>
> We use $[x]$ to denote the equivalence class of $x$.

<details>
<summary>Example of equivalence classes in path homotopy</summary>

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

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$X=\{p,q\}$ 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 $\mathbb{R}^n$.
</details>