Retraction: Difference between revisions

Latest revision as of 19:57, 11 May 2008

This article defines a property of continuous maps between topological spaces

Definition

Symbol-free definition

A retraction of a topological space is an idempotent continuous map from the topological space to itself. In other words, it is a continuous map from the topological space to a subspace, such that the restriction to that subspace is the identity map.

The image of a retraction is termed a retract (this is a subspace property).

Definition with symbols

Fill this in later

Facts

Given any functor from the category of topological spaces to the category of groups, or the category of modules over a ring, the topological space notion of retract gets mapped to the notion of retract in the relevant category.

Let $F$ be a functor from the category of topological spaces to the category of groups.

Suppose $X$ is a topological space, $r$ is a retraction and $Y$ is the image of the retraction. Suppose $i$ is the inclusion of $Y$ in $X$ . Since $r \circ i$ is the identity map on $Y$ , we get by functorality that $F (r) \circ F (i)$ is the identity on $F (Y)$ . This forces that $F (Y) \to F (X)$ is injective, and $F (X) \to F (Y)$ is surjective. Identifying $F (Y)$ with its image subgroup $F (X)$ , we can view $F (r)$ as a retraction of groups.

The above can be applied to the fundamental group, any of the higher homotopy groups, or any of the homology groups.

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 A '''retraction''' of a topological space is an idempotent [[continuous map]] from the topological space to itself. In other words, it is a continuous map from the topological space to a subspace, such that the restriction to that subspace is the identity map.
+The image of a retraction is termed a [[retract]] (this is a subspace property).
 ===Definition with symbols===
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 ==Facts==
-===Map of fundamental groups===
+Given any functor from the category of topological spaces to the category of groups, or the category of modules over a ring, the topological space notion of retract gets mapped to the notion of retract in the relevant category.
-If <math>X</math> is a topological space and <math>f</math> is a retraction from <math>X</math> whose image is a subspace <math>Y</math>, then <math>f</math> induces a map of the fundamental group <math>\pi_1(X) \to \pi_1(Y)</math>. In fact, this induced map is a [[retraction of groups]]. This follows from the fact that there is also a map <math>\pi_1(Y) \to \pi_1(X)</math> induced by inclusion, and that the composite of these maps is the identity on <math>\pi_1(Y)</math>.
+Let <math>F</math> be a functor from the category of topological spaces to the category of groups.
-In fact, the same can be said for any functor to groups.
+Suppose <math>X</math> is a topological space, <math>r</math> is a retraction and <math>Y</math> is the image of the retraction. Suppose <math>i</math> is the inclusion of <math>Y</math> in <math>X</math>. Since <math>r \circ i</math> is the identity map on <math>Y</math>, we get by functorality that <math>F(r) \circ F(i)</math> is the identity on <math>F(Y)</math>. This forces that <math>F(Y) \to F(X)</math> is injective, and <math>F(X) \to F(Y)</math> is surjective. Identifying <math>F(Y)</math> with its image subgroup <math>F(X)</math>, we can view <math>F(r)</math> as a [[retraction of groups]].
-Thus, if a subspace is a retract, then the mapping of fundamental groups from the subspace to the whole space is injective.
+The above can be applied to the [[fundamental group]], any of the higher [[homotopy group]]s, or any of the [[homology group]]s.