Homotopy: Difference between revisions

Revision as of 02:47, 9 November 2010

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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 ==Definition==
-Suppose <math>X, Y</math> are [[topological space]]s and <math>f,g:X \to Y</math> are [[continuous map]]s from <math>X</math> to <math>Y</math>. A continuous map <math>F: X \times I \to Y</math> is termed a '''homotopy''' from <math>f</math> to <math>g</math> if for every <math>x \in X</math>, <math>F(x,0) = f(x)</math> and <math>F(x,1) = g(x)</math>.
+===A homotopy that takes time <math>1</math>===
+Suppose <math>X, Y</math> are [[topological space]]s and <math>f,g:X \to Y</math> are [[continuous map]]s from <math>X</math> to <math>Y</math>. Let <math>I</math> be the [[defining ingredient::closed unit interval]] <math>[0,1]</math>.
+A continuous map <math>F: X \times I \to Y</math> is termed a '''homotopy''' from <math>f</math> to <math>g</math> if for every <math>x \in X</math>, <math>F(x,0) = f(x)</math> and <math>F(x,1) = g(x)</math>.
+Note that <math>F</math> has to be a continuous map from <math>X \times I</math> equipped with the [[product topology]]. It is ''not'' sufficient to require that <math>F</math> be a [[separately continuous map]] in each coordinate, i.e., it is not enough to insist that <math>x \mapsto F(x,t)</math> is continuous for each <math>t</math> and <math>t \mapsto F(x,t)</math> is continuous for each <math>x</math>.
+===A homotopy that takes time <math>T > 0</math>===
+Suppose <math>X, Y</math> are [[topological space]]s and <math>f,g:X \to Y</math> are [[continuous map]]s from <math>X</math> to <math>Y</math>. A homotopy from <math>f</math> to <math>g</math> that takes time <math>T</math> is a continuous map <math>F: X \times [0,T] \to Y</math> such that <math>F(x,0) = f(x)</math> and <math>F(x,T) = g(x)</math> for all <math>x</math> in <math>X</math>.
+Given any homotopy that takes time <math>T</math>, there is a linear scaling of the homotopy to a homotopy that takes time <math>1</math>, which would make it a homotopy in the first sense. The main advantage of considering homotopies that take time <math>T</math> is that these have an associative multiplication.
 ==Related notions==