Homotopies

Overview

Intuitively, two maps are homotopic if one can be continuously deformed into the other. Path-homotopies are paths that can be deformed into each other when endpoints are fixed.

Path-homotopy classes are also elements of the fundamental groups of a space.

See also: Path concatenation, Homotopy equivalence

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Ways to (dis-)prove homotopies

• (Path-)connectedness (ex: antipodal maps on the real line with origin removed)

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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 $f\simeq f’$.

Alternatively, a homotopy can be viewed as a one-parameter family $t\mapsto H_t$. 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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Homotopies relative to a subspace

Definition: Homotopy relative to a subspace

Given a subspace $A\subseteq X$, a homotopy $F:X\times I\to Y$ is called a homotopy relative to $A$ (or homotopy rel $A$) if values of $F$ on $A$ are independent of the time parameter, i.e., for all $a\in A$, we have $F(a,t)=F(a,0)$ for all $t\in I$. Equivalently, the restriction $F_t|A$ is the same for all $t\in I$.

We may also say that functions $f,g:X\to Y$ are homotopic rel $A$ if there exists a homotopy $F$ between them with the above properties.

Note that a deformation retraction of $X$ onto $A\subseteq X$ is precisely a homotopy rel $A$ from the identity of $X$ to a retraction of $X$ onto $A$.

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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

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

Maps on the real plane with origin removed

Let $X={\mathbb{R}}^2\backslash \{0\}$ be the real plane with origin removed. Then

$$f_0(x)=x{f}_1(x)=-x$$

are homotopic via

$$H(x,t)=\left(\begin{array}{cc}\cos \pi t& -\sin \pi t\\ \sin \pi t& \cos \pi t\end{array}\right)x,$$

which continuously rotates the point. This is well-defined (i.e., $H(x,t)\in X$ for all $x\in X$), since we have $\det =1$, hence the matrix is invertible and $H(x,t)$ is always nonzero.

Note that these are not homotopic if replacing the plane with $\mathbb{R}\backslash \{0\}$ (argue based on (Path-)connectedness) or ${\mathbb{R}}^3\backslash \{0\}$ (argument TBD).