Homotopies and path concatenation

Overview §

See also: (Theorem) Path concatenation on path-homotopy classes satisfies group axioms

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(Path-)homotopies and (path-)homotopy classes §

Definition: Homotopic, homotopy

Let $I=[0,1]$, $X,Y$ be topological spaces, and $f,f’:X\to Y$ be continuous. We say $f$ is homotopic to $f’$ if there exists a continuous mapping $H:X\times I\to Y$ such that for all $x\in X$, $H(x,0)=f(x)$ and $H(x,1)=f’(x)$. We call $H$ a homotopy from $f$ to $f’$, and we write $f\simeq f’$.

As an alternative perspective, for every $t\in [0,1]$, define a continuous $H_t:X\to Y$ by $H_t(x)=H(x,t)$ such that $H_0=f$, $H_1=f’$, and the $H_t$s vary continuously with respect to $t$ (when you put them all together, $H$ is continuous).

Definition: Path-homotopy

Given $I=[0,1$, two paths $f,f’:I\to X$, e.g., from $x_0\in X$ to $x_1\in X$, are path-homotopic, written $f{\simeq }_{p}f’$ if there exists a homotopy $H:I\times I\to X$ such that $H(0,t)=x_0$ and $H(1,t)=x_1$; that is, the endpoints are fixed for all $t\in I$.

Lemma: Homotopies and path-homotopies are equivalence relations.

Proof from Topology.

We check the three properties of an equivalence relation for homotopies:

• Reflexivity: For $f:X\to Y$, we have $f\simeq f$ via $H:X\times I\to Y$ defined by $H(x,t)=f(x)$. Note that $H=f\circ {\pi }_1$, the projection onto the first coordinate.
• Symmetry: If $f\simeq g$ via $H:X\times I\to Y$, then we have $g\simeq f$ via $H’:X\times I\to Y$ defined by $H’(x,t)=H(x,1-t)$.
• Transitivity: If $f\simeq f’$ via $H$ and $f’\simeq f’’$ via $H’$, define $H’’:X\to Y$ by

$$H^{\prime \prime }(t)=\{\begin{array}{ll}H(x,2t)& \text{if }0\le t\le 1/2\\ H^{\prime }(x,2t-1)& \text{if }1/2\le t\le 1.\end{array}$$

$H’’$ is well-defined (hint: how is it defined at $t=1/2$?) and continuous on the closed sets $X\times [0,1/2]$ and $X\times [1/2,1]$, which implies $H’’$ is continuous by the pasting lemma

🔺 The same homotopies $H$, etc. work for showing path-homotopy, with the additional step of checking the endpoint condition. $\square$

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Path concatenation §

Definition: Path concatenation

If $f$ is a path in $X$ from $x_0$ to $x_1$, and $g$ is a path in $X$ from $x_1$ to $x_2$, their concatenation or composition is a path from $x_0$ to $x_2$ defined by

$$(f\ast g)(s)=\{\begin{array}{ll}f(2s)& 0\le s\le 1/2\\ g(2s-1)& 1/2\le s\le 1.\end{array}$$

The operation $\ast$ induces a well-defined operation on path-homotopy classes, which is given by

$$[f]\ast [g]=[f\ast g].$$

Lemma: Concatenation properties with path homotopies

Let $f,f^{\prime },g:I\to X$ be paths.

• (i) If $f{\simeq }_{p}f’$ and $g{\simeq }_{p}f’$, then $f\ast g{\simeq }_{p}f’\ast g’$.
• (ii) If $h:X\to Y$ is continuous and $f{\simeq }_{p}f’$, then $h\circ f{\simeq }_{p}h\circ f’$.
• (iii) $h\circ (f\ast g)=(h\circ f)\ast (h\circ g)$.

Proof from Topology.

• (i) Let $H$ be a path-homotopy from $f$ to $f’$ and $G$ be a path-homotopy from $g\to g’$. Define

$$F(s,t)=\{\begin{array}{ll}H(2s,t)& 0\le s\le 1/2\\ G(2s-1,t)& 1/2\le s\le 1.\end{array}$$

This is well-defined since $H(1,t)=x_1=G(0,t)$, and continuous by the pasting lemma. 🔺Check that $F$ is a path-homotopy from $f\ast g$ to $f’\ast g’$.

• (ii) 🔺 Check that if $H$ is a path-homotopy from $f$ to $f$, then $h\circ H$ is a path-homotopy from $h\circ f$ to $h\circ f’$.
• (iii) 🔺 Clear from the definition: just plug in values of $t$…or $s$? $\square$

Transclude of (Theorem)-Path-concatenation-on-path-homotopy-classes-satisfies-group-axioms#^27df0f

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Examples §

Straight line homotopy in convex sets §

Definition: Convex set

A subset $A\subseteq {\mathbb{R}}^n$ is convex if for all $x,y\in A$, the line segment from $x$ to $y$ lies in $A$. Explicitly, the entire line $(1-t)x+ty$ is in $A$ for all $t\in [0,1]$.

If $A\subseteq \mathbb{R}$ is convex, then any two continuous functions $f,f’:X\to A$ are homotopic via the straight line homotopy

$$H(x,t)=(1-t)f(x)+t(f^{\prime }(x)).$$

❓ What is $X$?