Suspension: Difference between revisions

Latest revision as of 02:58, 25 December 2010

This article describes a construct that involves some variant of taking a product of a topological space with the unit interval and then making some identifications, typically at the endpoints, based on some specific maps.
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Definition

Long definition

Given a topological space $X$ , the suspension of $X$ , denoted $S X$ , is defined as the quotient of $X \times I$ (where $I$ is the closed unit interval and we use the product topology) by the following two equivalence relations:

$(x_{1}, 0) \sim (x_{2}, 0), \forall x_{1}, x_{2} \in X$

and

$(x_{1}, 1) \sim (x_{2}, 1) \forall x_{1}, x_{2} \in X$

Short definition

The suspension $S X$ of a topological space $X$ can be described in the following succinct ways as a quotient space $S X = (X \times [0, 1] / (X \times {0})) / (X \times {1})$ . In other words, we quotient out successively (or simultaneously) by the subspaces $X \times {0}$ and $X \times {1}$ .

Related constructs

In terms of other constructions

Double mapping cylinder

The suspension can be viewed as a case of a double mapping cylinder where $Y$ and $Z$ are both one-point spaces and both the maps involved send $X$ to the one point.

Join

The suspension can also be viewed as the join of $X$ with the 0-sphere $S^{0}$ .

Relation between a space and its suspension

Homology for suspension

Further information: homology for suspension

Taking the suspension shifts the homology groups. Specifically, for any topological space $X$ :

$H_{k + 1} (S X) ≅ H_{k} (X), k \geq 1$

The result extends to the zeroth homology if we use reduced homology instead of homology. (Without reduced homology, the formulation becomes more clumsy):

${\tilde{H}}_{k + 1} (S X) ≅ {\tilde{H}}_{k} (X), k \geq 0$

This result is an easy application of the Mayer-Vietoris homology sequence, and is similar to the application of the Seifert-van Kampen theorem to show that suspension of path-connected space is simply connected.

@@ Line 1: / Line 1: @@
+{{interval-cum-mapping construct}}
 ==Definition==
-Given a topological space <math>X</math>, the suspension of <math>X</math>, denoted <math>SX</math>, is defined as the quotient of <math>X \times I</math> by the following two equivalence relations:
+===Long definition===
-<math>(x_1,0) \sim (x_2,0)</math>
+Given a topological space <math>X</math>, the suspension of <math>X</math>, denoted <math>SX</math>, is defined as the [[defing ingredient::quotient topology|quotient]] of <math>X \times I</math> (where <math>I</math> is the [[defining ingredient::closed unit interval]] and we use the [[defining ingredient::product topology]]) by the following two equivalence relations:
+<math>\! (x_1,0) \sim (x_2,0), \forall \ x_1,x_2 \in X</math>
 and
-<math>(x_1,1) \sim (x_2,1)</math>
+<math>\! (x_1,1) \sim (x_2,1) \ \forall \ x_1,x_2 \in X</math>
-Also see:
+===Short definition===
+The suspension <math>SX</math> of a topological space <math>X</math> can be described in the following succinct ways as a [[quotient space]] <math>SX = (X \times [0,1]/(X \times \{ 0 \}))/(X \times \{ 1 \})</math>. In other words, we quotient out successively (or simultaneously) by the subspaces <math>X \times \{ 0 \}</math> and <math>X \times \{ 1 \}</math>.
+==Related constructs==
 * [[Suspension functor]]
@@ Line 29: / Line 37: @@
 {{further|[[homology for suspension]]}}
+Taking the suspension shifts the homology groups. Specifically, for any [[topological space]] <math>X</math>:
+<math>H_{k + 1}(SX) \cong H_k(X), \qquad k \ge 1</math>
+The result extends to the zeroth homology if we use reduced homology instead of homology. (Without reduced homology, the formulation becomes more clumsy):
+<math>\tilde{H}_{k+1}(SX) \cong \tilde{H}_k(X), \qquad k \ge 0</math>
+This result is an easy application of the [[Mayer-Vietoris homology sequence]], and is similar to the application of the [[Seifert-van Kampen theorem]] to show that [[suspension of path-connected space is simply connected]].