Homotopy

Formal Definition

A Homotopy is a continuous deformation between two continuous maps.

Let $f,g:X\to Y$ be continuous maps. We say that $f$ and $g$ are homotopic, written

$$f\simeq g,$$

if there is a family of maps $h_t:X\to Y$ for $t\in I=[0,1]$ such that:

• $h_0=f$ and $h_1=g$,
• the map $H:X\times I\to Y$ defined by $H(x,t)=h_t(x)$ is continuous.

The map $H$ is called a homotopy from $f$ to $g$. The continuity of $H$ implies in particular that each individual map $h_t:X\to Y$ is continuous. homotopy is an equivalence relation on maps $X\to Y$

nullhomotopic

A constant map is a $c_{x_0}:X\to X,x\mapsto x_0$ A map $f:X\to Y$ is nullhomotopic if it is homotopic to a constant map, meaning that there exists a homotopy $H:X\times I\to X$ such that $H(x,0)=f(x)$ and $H(x,1)=x_0$

Homotopy Equivalence

A map $f:X\to Y$ is a homotopy equivalence if there exists a map $g:Y\to X$ such that

$$g\circ f\simeq 1_X\text{and}f\circ g\simeq 1_Y.$$

In this case, $X$ and $Y$ are called homotopy equivalent, or said to have the same homotopy type.

Homotopy type is an equivalence relation:

• Reflexive: $1_X:X\to X$ is a homotopy equivalence.
• Symmetric: if $f:X\to Y$ is a homotopy equivalence with inverse up to homotopy $g:Y\to X$, then $g$ is also a homotopy equivalence.
• Transitive: the composition of homotopy equivalences is again a homotopy equivalence.

Homotopy Class

The equivalence class of $f:X\to Y$ is called its homotopy class and denoted $[f]$

The set of homotopy classes between two spaces $X$ and $Y$ is usually written as $[X,Y]$ Its elements are not spaces, but equivalence classes of maps $f:X\to Y$, where two maps are considered the same if they are homotopic.

Contractible Spaces

A space $X$ is contractible if it has the homotopy type of a point.

$$X\text{ is contractible}⟺1_X\text{ is nullhomotopic}.$$

note that $1_X$ is the identity map of a space to it self: $1_X:X\to X$ with $x\mapsto x$

Example: ${\mathbb{R}}^n$

The Euclidean space ${\mathbb{R}}^n$ is homotopy equivalent to the one-point space $\{\ast \}$

Let $f:{\mathbb{R}}^n\to \{\ast \}$ be the unique map, and let $g:\{\ast \}\to {\mathbb{R}}^n$ send $\ast$ to the origin. We want to show that $f\circ g\simeq 1_{\{\ast \}}$ and $g\circ f\simeq 1_{{\mathbb{R}}^n}$ We know $f\circ g=1_{\{\ast \}}⟹f\circ g\simeq 1_{\{\ast \}}$ $g\circ f:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is the constant map sending every point to $0$. The homotopy

$$H:{\mathbb{R}}^n\times I\to {\mathbb{R}}^n,H(x,t)=tx$$

deforms the constant map $H(x,0)=0$ to the identity map $H(x,1)=x$. Hence

$$g\circ f\simeq 1_{{\mathbb{R}}^n},$$

so ${\mathbb{R}}^n$ is contractible.

Deformation Retractions

A deformation retraction of $X$ onto a subspace $A\subset X$ is a homotopy $f_t:X\to X$ such that:

• $f_0=1_X$,
• $f_t{|}_A=1_A$ for all $t\in I$,
• $f_1(X)=A$.

Thus every point of $X$ is continuously moved into $A$, while every point of $A$ stays fixed throughout the homotopy.

In this case, $f_1:X\to A$ is a homotopy equivalence with homotopy inverse given by the inclusion

$$A\hookrightarrow X.$$

Therefore, if $A$ is a deformation retract of $X$, then $X$ and $A$ have the same homotopy type.

Homotopy of path

A homotopy of paths in $X$ is a family $f_t:I\to X,0\le t\le 1$ , such that

• $f_t(0)=x_0$ and $f_t(1)=x_1$ (independent of $t$)
• The associated map $F:I\times I\to X$ defined by $F(s,t)=f_t(s)$ is continuous

product path

The composition or product path $f\cdot g$ of two paths is defined by the formula

$$f\cdot g(s)=\{\begin{array}{l}f(2s),0\le s\le \frac{1}{2}\\ g(2s-1),\frac{1}{2}\le s\le 1\end{array}$$

Geometric Intuition: Blender :D

In Blender (the software), one can create a smooth spine curve $C$ and use Geometry Nodes to generate a 3D tubular shape around it. One can then use a parameter $t\in [0,1]$ to control the tube’s radius so it changes continuously over time to make an animation ($t$ change continuously).

• Consider the solid tube $V_t=\{x:dist(x,C)\le r(t)\}$
  • Each map from $V_t$ to $C$ is a deformation retraction by nearest-point projection.
• Consider the surface of the tube $S_t=\{x\in {\mathbb{R}}^3:dist(x,C)=r(t)\}$
  • The changing $t$ produces a homotopy