📂Topological Data Analysis

Homotopy Classes

Theorem

Brief Description

In any topological space, the relation of a homotopy defined between any two fixed points is an equivalence relation.

Detailed Description

Given a topological space $X$ and two points $x_{0}, x_{1} \in X$, if the paths $f, g : I \to X$ between two points are homotopic, as expressed by $f \simeq g$, then this binary relation $\simeq$ is an equivalence relation. Moreover, the equivalence classes created by this equivalence relation $\simeq$ $\left\{ g : f \simeq g \right\}$ are represented as $[f]$.

Explanation

At first glance, this theorem may be misunderstood to mean that all paths in the space $X$ with given points $x_{0}, x_{1}$ are represented only by the two points. However, that is only the case when a homotopy for all paths exists, and as a simple example, considering a torus reveals that with a hole in the middle of the space, not all paths can have a homotopy.

Fortunately, it holds in a general Euclidean space $\mathbb{R}^{p}$, and more generally, one can conjecture that it would hold in a convex vector space as well.

Proof ¹

To show that $\simeq$ is reflective, symmetric, and transitive, Reflexivity is trivial since there exists a constant homotopy $\left\{ h_{t} = f \right\}$ between $f \simeq f$. Symmetry is also trivial since for $h_{t}$ existing between $f$ and $g$, $\left\{ h_{1-t} \right\}$ exists as a homotopy between $g$ and $f$. Transitivity is a bit more complex. When the binary continuous function corresponding to path $f : I \to X$ is $$F : I \times I \to X$$ and the binary continuous function corresponding to path $g : I \to X$ is $$G : I \times I \to X$$ if the binary continuous function corresponding to a mediating path $h$ is defined as $$H (s,t) = \begin{cases} F \left( s, 2t \right) & , \text{if } t \in [0,1/2] \\ G \left( s, 2t - 1 \right) & , \text{if } t \in [1/2,1] \end{cases}$$ then one can directly verify that a homotopy that makes $f \simeq g$ possible exists if $f \simeq h$ and $h \simeq g$, which visually appears as if the domains of the two functions defined from $I^{2}$ are halved and then connected.

The discussion on whether the mentioned $h_{t}$ are well-defined as functions and truly continuous is omitted here.

■