📂Topological Data Analysis

Definition of Homotopy

Definitions ¹

Let’s assume that the closed unit interval $I := [0,1]$ and the topological space $X$ are given.

1. A continuous function $p : I \to X$ from $x_{0}$ to $x_{1}$ satisfying the following for fixed points $x_{0} , x_{1} \in X$ is called a path or path. $$ \begin{align*} p(0) =& x_{0} \\ p(1) =& x_{1} \end{align*} $$
2. For two paths $f \equiv h_{0}$ and $g \equiv h_{1}$, the set $\left\{ h_{t} \right\}_{t \in [0,1]}$ of paths $h_{t} : I \to X$ satisfying the following two conditions is called a Homotopy:
   • (i): Independent of $t$, and $h_{t} (1) = x_{1}$.
   • (ii): For all $s,t \in I$, $H : I \times I \to X$ defined as $H(s,t) := h_{t} (s)$ is continuous. Depending on the context, either $H$ or $h_{t}$ itself is called a homotopy.
3. When there exists a homotopy between two paths $f$ and $g$, it is said that $f$ and $g$ are Homotopic, which is denoted as $f \simeq g$.

Description

Homotopy, simply put, refers to the functions that continuously join two given points, and the reason for considering this concept is to treat them mathematically as identical if the ways to connect two points are essentially the same. For example, in the illustration above, there are countless ways to join the point on the left with the point on the right, but in the aspect of (known to the public) topology, what does it matter whether you go straight or slightly sideways? [ NOTE: Here, the expression ‘slightly’ means the ‘continuity’ mentioned in the definition of $H$. ]

The diagram above shows that $H$ represents the function $h_{t}$ changing continuously according to $s$ in the unit square $I^{2}$.

The Meaning of Homotopy

As mentioned before, the idea that two paths are homotopic can be understood to mean that by slightly altering one path, it can be turned into the other. However, in topology, exploring such ‘practical sameness’ is to see the ’true difference’.

For example, consider two paths connecting two points on a torus as shown above. Since a torus has a hole in the middle, there is no homotopy joining the blue path and the red path, making them not homotopic. It’s important to see that by examining not just ‘points to points’ but ’the relations between points,’ we reach a stage where we can classify torus from non-torus convex shapes. Put grandly, studying homotopy is not merely a jargon-filled endeavor understandable only to non-specialists but a fresh methodology for viewing the essence of space.

Equivalence Condition

It’s not something to prove but just to mention passingly, but practically, the following definition might be more convenient to use.

$f$ and $g$ being homotopic is equivalent to the existence of a continuous function $H : I \times I \to X$ satisfying the following two conditions:

• (i): For all $t \in I$, $H(0,t) = x_{0}$ and $H(1,t) = x_{1}$.
• (ii): For all $s \in I$, $H (s, 0) = f(s)$ and $H(s,1) = g(s)$.

Homotopy as Path of Paths

• The following discussion might be a bit complex, so feel free to skip if it seems too daunting.

$$ \begin{align*} h_{t} (s) =& x_{0} \to x_{1} & \text{ as } s = 0 \to 1 \\ h_{t} = & f \to g & \text{ as } t = 0 \to 1 \end{align*} \qquad \cdots 🤔 ! $$

Formally, a path $f,g : I \to X$ in $X$ connecting two points $x_{0}, x_{1}$ is an element $f,g \in C \left( I, X \right)$ of the space of continuous functions, and $h_{t} : I \to C \left( I , X \right)$ is a path between two paths $f, g$ in that function space. $$ h_{t} \in C \left( I , C \left( I , X \right) \right) $$ There’s no reason not to call it a path. The reason we don’t use such expressions explicitly in the definition is that mentioning the natural topology of $C \left( I , X \right)$ as a topological space just for the definition of homotopy is too much. After all, definitions are better when shorter, and to discuss homotopy, requiring the continuity of the bivariate function $H$ is sufficient.

Talking about the continuity in a function space rather than $H$’s continuity would first require a topology on the function space, like the compact-open topology of a function space, which is overkill. Of course, just because the definition appears simple in the literature, we don’t necessarily need to stop at just knowing the definition. Homotopy has been simply defined above, but let’s take a moment with the remaining time to imagine beyond that. In mathematics, the function $$ F : Y \to X $$ of any set $Y, X$ cannot but garner interest. Thinking about a function of functions, that is, a new function $$ H : Z \to X^{Y} \iff H : Z \times Y \to X $$ with domain $Z$ and codomain the function space $X^{Y}$, endlessly brings new research topics like sequences of functions or inner spaces. If you agree that such curiosity is natural, now imagine inserting the closed unit interval $I = [0,1]$ in place of $Z$ and $Y$. $$ H : I \times I \to X $$

This is nothing but what we saw in the definition of $H$. In other words, despite its complicated name, homotopy is merely

• defined from $I \times I$ and involves
• an area of obvious interest:
• continuous functions of continuous functions where
• only the starting and ending points are specified
• among a set of functions

Not more than that. If homotopy seems unfamiliar and unintentionally large, be grateful for staying within the tiny scope of $I \times I$ instead of the vast expanse of $Y,Z$.