Smash product - Revision history

Vipul: 5 revisions

2008-05-11T19:58:32Z

5 revisions

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Vipul: /* Particular cases */

2007-09-26T21:58:33Z

Particular cases

← Older revision		Revision as of 21:58, 26 September 2007
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	The reduced suspension of a topological space is its smash product with the [[based circle]] (since the circle is homogeneous, it does not matter what basepoint we choose for it. The reduced suspension is often just called the ''suspension''.		The reduced suspension of a topological space is its smash product with the [[based circle]] (since the circle is homogeneous, it does not matter what basepoint we choose for it. The reduced suspension is often just called the ''suspension''.

			===Spheres===

			The <math>n</math>-sphere (with basepoint) can be viewed as the smash product of the (based) circle with itself <math>n</math> times.

Vipul at 21:37, 26 September 2007

2007-09-26T21:37:12Z

← Older revision		Revision as of 21:37, 26 September 2007
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	{{product ~~notion for spaces~~}}		{{product in the category\|based topological space}}

	==Definition==		==Definition==

Vipul at 21:36, 26 September 2007

2007-09-26T21:36:07Z

← Older revision		Revision as of 21:36, 26 September 2007
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	==Definition==		==Definition==

	Given two topological ~~spaces with basepoint,~~ <math>(X,x_0)</math> and <math>(Y,y_0)</math>, their '''smash product''' is defined as the quotient of <math>X \times Y</math> by the following equivalence relation:		===Naive definition===
			Given two [[based topological space]]s <math>(X,x_0)</math> and <math>(Y,y_0)</math>, their '''smash product''' denoted <math>(X \wedge Y,*)</math> is defined as the quotient of <math>X \times Y</math> by the following equivalence relation:

	<math>(x,y_0) \simeq (x_0,y)</math>		<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 (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).		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).

			===Category-theoretic definition===

			The smash product is the natural notion of product in the category of [[based topological space]]s. Category-theoretically, this means that given maps of based topological spaces to <math>X</math> and <math>Y</math> from <math>Z</math>, there exists a unique map to <math>X \wedge Y</math> from <math>Z</math> such that the diagram commutes.

	==Particular cases==		==Particular cases==

	{{~~fillin~~}}		===Reduced suspension===

			{{further\|[[reduced suspension]]}}

			The reduced suspension of a topological space is its smash product with the [[based circle]] (since the circle is homogeneous, it does not matter what basepoint we choose for it. The reduced suspension is often just called the ''suspension''.

Vipul at 14:00, 22 May 2007

2007-05-22T14:00:57Z

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	<math>(x,y_0) \simeq (x_0,y)</math>		<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==		==Particular cases==

	{{fillin}}		{{fillin}}

Vipul at 13:33, 22 May 2007

2007-05-22T13:33:29Z

New page

{{product notion for spaces}}

==Definition==

Given two topological spaces with basepoint, <math>(X,x_0)</math> and <math>(Y,y_0)</math>, their '''smash product''' is defined as the quotient of <math>X \times Y</math> by the following equivalence relation:

<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.

==Particular cases==

{{fillin}}