Homotopy

Definition

We begin by defining homotopies that take time $1$, but in the last subsection consider a variant notion of a homotopy that could take time $T>0$.

Definition as a jointly continuous map from the product with the unit interval

Suppose $X,Y$ are topological spaces and $f,g:X\to Y$ are continuous maps from $X$ to $Y$. Let $I$ be the closed unit interval $[0,1]$.

A jointly continuous map $F:X\times I\to Y$ is termed a homotopy from $f$ to $g$ if for every $x\in X$, $F(x,0)=f(x)$ and $F(x,1)=g(x)$.

Note that $F$ has to be a continuous map from $X\times I$ equipped with the product topology. It is not sufficient to require that $F$ be a separately continuous map in each coordinate, i.e., it is not enough to insist that $x\mapsto F(x,t)$ is continuous for each $t$ and $t\mapsto F(x,t)$ is continuous for each $x$.

Definition as a path in a function space

This definition works (at least) in the case that both $X$ and $Y$ are compactly generated Hausdorff spaces (probably in more cases). Under this definition, a homotopy between two continuous maps $f,g:X\to Y$ is a path from the point $f$ to the point $g$ in the topological space $C(X,Y)$ defined as the set of continuous maps from $X$ to $Y$ equipped with the compact-open topology.

Equivalence of definitions

A map $F:X\times [0,1]\to Y$ is equivalent to a map $\gamma _{F}$ from $[0,1]$ to the space $Y^{X}$ of functions from $X$ to $Y$, via the following rule:

$\!\gamma _{F}(t):=x\mapsto F(x,t)$

and in reverse:

$\!F_{\gamma }(x,t):=(\gamma (t))(x)$

It is further true that if $F$ is jointly continuous, then for each $t\in [0,1]$, $\gamma _{F}(t)$ is a continuous map. Thus, the homotopy from $f$ to $g$ is a map $\gamma$ from $[0,1]$ to the set $C(X,Y)$ of all continuous maps from $X$ to $Y$, where $\gamma _{R}(0)=f$ and $\gamma _{F}(1)=g$.

However, any set map from $[0,1]$ to $C(X,Y)$ need not be a homotopy, because the corresponding map $F$ need not be jointly continuous. It turns out that when both $X$ and $Y$ are compactly generated Hausdorff spaces, then a map $\gamma :[0,1]\to C(X,Y)$ is a continuous map to $C(X,Y)$ equipped with the compact-open topology iff the corresponding $F_{\gamma }$ is a jointly continuous map.

Variant: a homotopy that takes time $T>0$

Suppose $X,Y$ are topological spaces and $f,g:X\to Y$ are continuous maps from $X$ to $Y$. A homotopy from $f$ to $g$ that takes time $T$ is a continuous map $F:X\times [0,T]\to Y$ such that $F(x,0)=f(x)$ and $F(x,T)=g(x)$ for all $x$ in $X$.

Given any homotopy that takes time $T$, there is a linear scaling of the homotopy to a homotopy that takes time $1$, which would make it a homotopy in the first sense. The main advantage of considering homotopies that take time $T$ is that these have an associative multiplication.

Related notions

• Self-homotopy. Also check out Category:Properties of self-homotopies
• Smooth homotopy and piecewise smooth homotopy
• Linear homotopy and piecewise linear homotopy

Facts

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