Smooth Homotopy is an Equivalence Relation

Theorem
Let $X$ and $Y$ be smooth manifolds.

Let $K \subseteq X$ be a (possibly empty) subset of $X$.

Let $\map {\CC^\infty} {X, Y}$ be the set of all smooth mappings from $X$ to $Y$.

Define a relation $\sim$ on $\map \CC {X, Y}$ by $f \sim g$ if $f$ and $g$ are smoothly homotopic relative to $K$.

Then $\sim$ is an equivalence relation.

Proof
We examine each condition for equivalence.

Reflexivity
For any function $f: X \to Y$, define $F: X \times \closedint 0 1 \to Y$ as:
 * $\forall \tuple {x, t} \in X \times \closedint 0 1: \map F {x, t} = \map f x$

This yields a smooth homotopy between $f$ and itself.

So $\sim$ has been shown to be reflexive.

Symmetry
Given a homotopy:
 * $F: X \times \closedint 0 1 \to Y$ from $\map f x = \map F {x, 0}$ to $\map g x = \map F {x, 1}$

the mapping:
 * $\map G {x, t} = \map F {x, 1 - t}$

is a homotopy from $g$ to $f$.

Thus $G$ is smooth whenever $F$ is.

So $\sim$ has been shown to be symmetric.

Transitivity
The continuous case admits of a simpler solution than the smooth case, but the smooth case implies the continuous case, so we examine only the smooth case.

Define the function:
 * $\map \beta x = \begin{cases}

e^{-1 / \paren {1 - x^2} } & : \size x < 1 \\ 0 & : \text { otherwise} \end{cases}$

This function is known to be smooth.

Define:
 * $\ds \map \phi t = \dfrac {\ds \int_0^t \map \beta {\dfrac {x + 2} 4} \rd x} {\ds \int_0^1 \map \beta {\dfrac {x + 2} 4} \rd x}$

By construction, $\phi$ is a smooth function which is $0$ for all $t \le 1/4$, $1$ for all $t \ge 3/4$, and rises smoothly from $0$ to $1$ in $\closedint {\dfrac 1 4} {\dfrac 3 4}$.

Let $f, g, h$ be smooth functions such that $f$ is homotopic to $g$, which is in turn homotopic to $h$.

Then we can define the smooth homotopies:
 * $\map A {x, t} = \map \phi t \map g x + \paren {1 - \map \phi t} \map f x$, which satisfies $\map A {x, 0} = \map f x$ and $\map A {x, 1} = \map g x$

and:
 * $\map B {x, t} = \map \phi t \map h x + \paren {1 - \map \phi t} \map g x$, which satisfies $\map B {x, 0} = \map g x$ and $\map B {x, 1} = \map h x$

We then define a smooth function:


 * $\map C {x, t} = \begin {cases} \map A {x, \dfrac t 2} & : t \le \dfrac 1 2 \\

\map B {x, \dfrac {t + 1} 2} & : t > \dfrac 1 2 \end {cases}$

$C$ is a smooth function satisfying $\map C {x, 0} = \map f x$ and $\map C {x, 1} = \map h x$, so it is a homotopy from $f$ to $h$.

So $\sim$ has been shown to be transitive.

$\sim$ has been shown to be reflexive, symmetric and transitive.

Hence by definition it is an equivalence relation.