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> [!definition] Definition. ([[homotopy]])
> Let $X,Y$ be [[topological space|topological spaces]]. We say two [[continuous]] maps $f:X \to Y$ and $g:X \to Y$ are (freely) **homotopic** if there exists a [[continuous]] map $H: X \times I \to Y$ s.t. $H(x,0)=f(x) \text{ and } H(x,1)=g(x)$
for each $x \in X$ (here, $I=[0,1]$). The map $F$ is called a **homotopy** between $f$ and $g$. We write $f \simeq g$.
>
If $f$ is homotopic to $g$ and $g$ is a constant map, we say that $f$ is **nulhomotopic**.
\
For spaces $X$ and $Y$ and a [[subspace topology|subspace]] $A \subset X$, the [[relation]] "homotopic relative to $A
quot; on the set of maps from $X$ to $Y$ is an [[equivalence relation]]. The [[category]] $\mathsf{Htpy}$ has topological spaces as objects (just as $\mathsf{Top}$ does), and homotopy classes of maps as morphisms.
> [!intuition]
> Homotopy makes precise the notion of 'considering things (i.e., functions or spaces) *up to deformation*'.
^intuition
> [!basicexample] Example. (Non-identity-homotopic maps on the circle have fixed and anti-fixed point)
> Let $f:\mathbb{S}^{1} \to \mathbb{S}^{1}$ be a [[continuous]] map which is not [[homotopy|homotopic]] to the [[identity map]]. Then there exists an $x \in \mathbb{S}^{1}$ such that $f(x)=x$, and a $y \in \mathbb{S}^{1}$ such that $f(y)=-y$.
> \
> To see this, Contrapositively suppose $f$ has no fixed point, or $f$ has no anti-fixed point; first assume no anti-fixed point: $f(v) \neq -v$ for all $v \in \mathbb{S}^{1}$. We want to show $f$ is homotopic to $\id_{\mathbb{S}^{1}}$. Define a 'normalized [[straight-line homotopy]]' $\begin{align}
H: \mathbb{S}^{1} \times I \to & \mathbb{S}^{1} \\
(v,t) \mapsto & \frac{tv + (1-t)f(v) }{\|tv + (1-t)f(v)\|}
\end{align}.$
Clearly $H$ is [[continuous]], with $H(v,0)=f(v)$ and $H(v,1)=v=\id_{\mathbb{S}^{1}}(v)$. It is well-defined iff the denominator is always nonzero; the denominator is zero iff $tv+(1-t)f(v)=0 \iff v = \frac{t-1}{t}f(v).$
Since $\|v\|=1=\|f(v)\|$ and $t \in [0,1]$ (so $\frac{t-1}{t} \leq 0$) this condition holds iff $\frac{t-1}{t} = -1$. So $H$ is well-defined iff at $t=\frac{1}{2}$ we never have $v=-f(v)$. Of course, this holds by the assumption that $f$ has no anti-fixed points.
>
Next, instead suppose $f$ has no fixed point. This time we first define a [[homotopy]] $F$ between $f$ and $-\id_{\mathbb{S}^{1}}$ thus: $\begin{align}
F: \mathbb{S}^{1} \times I \to & \mathbb{S}^{1} \\
(v,t) \mapsto & \frac{-tv + (1-t)f(v) }{\|-tv + (1-t)f(v)\|}
\end{align}.$
It is immediate to verify using analogous procedures to the above that $F$ is [[continuous]], $F(v,0)=f(v)$, $F(v, 1)=-v$, and $F$ is [[well-defined]] iff $f$ has no fixed points. By [[antipodal map on the sphere is homotopic to the identity in odd dimensions]] we also have a [[homotopy]] $G$ from $-\id_{\mathbb{S}^{1}}$ to $\id_{\mathbb{S}^{1}}$. Because [[homotopy]] is an [[equivalence relation]], we conclude that since $f \cong -\id$ and $-\id \cong \id$ that $f \cong \id$ as required.
^basic-example
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####
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#### References
> [!backlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM [[]]
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```
> [!frontlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM outgoing([[]])
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```