Homotopy: Difference between revisions

Latest revision as of 23:28, 20 December 2010

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 $γ_{F}$ from $[0, 1]$ to the space $Y^{X}$ of functions from $X$ to $Y$ , via the following rule:

$γ_{F} (t) : = x \mapsto F (x, t)$

and in reverse:

$F_{γ} (x, t) : = (γ (t)) (x)$

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

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 $γ : [0, 1] \to C (X, Y)$ is a continuous map to $C (X, Y)$ equipped with the compact-open topology iff the corresponding $F_{γ}$ 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

Fill this in later

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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>.
+We begin by defining homotopies that ''take time <math>1</math>'', but in the last subsection consider a variant notion of a homotopy that could take time <math>T > 0</math>.
+===Definition as a jointly continuous map from the product with the unit interval===
+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 [[defining ingredient::jointly 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>.
+===Definition as a path in a function space===
+This definition works (at least) in the case that both <math>X</math> and <math>Y</math> are [[compactly generated Hausdorff space]]s (probably in more cases). Under this definition, a '''homotopy''' between two [[continuous map]]s <math>f,g:X \to Y</math> is a [[path]] from the point <math>f</math> to the point <math>g</math> in the [[topological space]] <math>C(X,Y)</math> defined as the set of continuous maps from <math>X</math> to <math>Y</math> equipped with the [[compact-open topology]].
+===Equivalence of definitions===
+A map <math>F: X \times [0,1] \to Y</math> is equivalent to  a map <math>\gamma_F</math> from <math>[0,1]</math> to the space <math>Y^X</math> of functions from <math>X</math> to <math>Y</math>, via the following rule:
+<math>\! \gamma_F(t) := x \mapsto F(x,t)</math>
+and in reverse:
+<math>\! F_\gamma(x,t) := (\gamma(t))(x)</math>
+It is further true that if <math>F</math> is jointly continuous, then for each <math>t \in [0,1]</math>, <math>\gamma_F(t)</math> is a continuous map. Thus, the homotopy from <math>f</math> to <math>g</math> is a map <math>\gamma</math> from <math>[0,1]</math> to the set <math>C(X,Y)</math> of all continuous maps from <math>X</math> to <math>Y</math>, where <math>\gamma_R(0) = f</math> and <math>\gamma_F(1) = g</math>.
+However, any set map from <math>[0,1]</math> to <math>C(X,Y)</math> need not be a homotopy, because the corresponding map <math>F</math> need not be jointly continuous. It turns out that when both <math>X</math> and <math>Y</math> are [[compactly generated Hausdorff space]]s, then a map <math>\gamma:[0,1] \to C(X,Y)</math> is a [[continuous map]] to <math>C(X,Y)</math> equipped with the [[compact-open topology]] iff the corresponding <math>F_\gamma</math> is a jointly continuous map.
+===Variant: 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==