Retraction - Revision history

Vipul: 4 revisions

2008-05-11T19:57:58Z

4 revisions

← Older revision	Revision as of 19:57, 11 May 2008
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Vipul: /* Facts */

2007-10-27T01:43:42Z

Facts

← Older revision		Revision as of 01:43, 27 October 2007
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	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]].		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]].

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

Vipul at 01:43, 27 October 2007

2007-10-27T01:43:03Z

← Older revision		Revision as of 01:43, 27 October 2007
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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.		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===		===Definition with symbols===
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	==Facts==		==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>~~.

	~~In fact, the same can~~ be ~~said for any~~ functor to groups.		Let <math>F</math> be a functor from the category of topological spaces to the category of groups.

	~~Thus~~, if a ~~subspace~~ is ~~a retract, then~~ the ~~mapping~~ of ~~fundamental groups from~~ the ~~subspace~~ to ~~the whole space~~ is injective.		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]].

Vipul at 01:36, 27 October 2007

2007-10-27T01:36:50Z

← Older revision		Revision as of 01:36, 27 October 2007
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	{{~~topospace~~ map property}}		{{continuous map property}}

	==Definition==		==Definition==
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	===Symbol-free definition===		===Symbol-free definition===

	A '''retraction''' of a topological space is a continuous ~~idempotent~~ 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.		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.

	===Definition with symbols===		===Definition with symbols===

Vipul at 03:05, 22 May 2007

2007-05-22T03:05:29Z

New page

{{topospace map property}}

==Definition==

===Symbol-free definition===

A '''retraction''' of a topological space is a continuous idempotent 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.

===Definition with symbols===

{{fillin}}

==Facts==

===Map of fundamental groups===

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>.

In fact, the same can be said for any functor to groups.

Thus, if a subspace is a retract, then the mapping of fundamental groups from the subspace to the whole space is injective.