Homotopy

Definition

A homotopy that takes time $1$

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 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 $x$.

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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