Suspension: Difference between revisions
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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: | 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: | ||
<math>(x_1,0) \sim (x_2,0)</math> | <math>\! (x_1,0) \sim (x_2,0)</math> | ||
and | and | ||
<math>(x_1,1) \sim (x_2,1)</math> | <math>\! (x_1,1) \sim (x_2,1)</math> | ||
Also see: | Also see: | ||
Revision as of 02:09, 7 October 2010
Definition
Given a topological space , the suspension of , denoted , is defined as the quotient of by the following two equivalence relations:
and
Also see:
In terms of other constructions
Double mapping cylinder
The suspension can be viewed as a case of a double mapping cylinder where and are both one-point spaces and both the maps involved send to the one point.
Join
The suspension can also be viewed as the join of with the 0-sphere .
Relation between a space and its suspension
Homology for suspension
Further information: homology for suspension