Smash product: Difference between revisions

Revision as of 14:00, 22 May 2007

Definition

Given two topological spaces with basepoint, $(X, x_{0})$ and $(Y, y_{0})$ , their smash product is defined as the quotient of $X \times Y$ by the following equivalence relation:

$(x, y_{0}) ≃ (x_{0}, y)$

In other words, we collapse both the copy of $X$ and the copy of $Y$ , through $(x_{0}, y_{0})$ , to a single point (the union of the $X$ -copy and $Y$ -copy is isomorphic to the wedge sum of the spaces, hence the smash product can be viewed as the quotient of the product by collapse of the wedge sum to a point).

Particular cases

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@@ Line 7: / Line 7: @@
 <math>(x,y_0) \simeq (x_0,y)</math>
-In other words, we collapse both the copy of <math>X</math> and the copy of <math>Y</math>, through <math>(x_0,y_0)</math>, to a single point.
+In other words, we collapse both the copy of <math>X</math> and the copy of <math>Y</math>, through <math>(x_0,y_0)</math>, to a single point (the union of the <math>X</math>-copy and <math>Y</math>-copy is isomorphic to the [[wedge sum]] of the spaces, hence the smash product can be viewed as the quotient of the product by collapse of the wedge sum to a point).
 ==Particular cases==
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