Understanding the Smash Product of Two Topological Spaces

[alyscia]

Hi all,

I do realize that my previous thread on CW complexes was unanswered, so perhaps I am posting my questions to wrong section of this forum. If so, please direct me to the right forum. Otherwise, I am having some problems understanding the smash product of two topological spaces. If anyone understands it well, could you give me a concrete (nontrivial) example of how a smash product works? (Wikipedia and Hatcher, which are my two primary sources of topology both do not seem to have a satisfactory example).

Thanks,

A

 

[eastside00_99]

What about the smash product of two circles? Consider the Cartesian product of two circles. Let x be a point in the first factor and y a point in the second factor. We naturally have

A=S^1\times {y}\subset S^1\times S^1 =T^1, B={x}\times S^1 \subset S^1\times S^1=T^1 .

A and B are what is known as pointed sets, they have a naturally defined topology on them, and can be identified as copies of S^1. In fact, the torus can be "broken up" into family of disjoint circles parametrized by the second factor S^1 (this is an example of a foliation). So, geometrically A is the circle which corresponds to the point y in S^1 (this is an example of a leaf of a foliation). If we think about it this way we can avoid using coordinates. Also T can also be thought of as a family of disjoint circles parametrized by the first factor S^1. So, B is the circle which corresponds to the point x in the first factor x in S^1. In other words, we have two circles on the torus which intersect each other at exactly one point.

Wikipedia actually has a sorry article because the identification is not worded well. What it really means is that we identify all the points in A and all the points in B as being equivalent. The most natural thing for us to do is to identify all the points in A and all the points in B as being the point (x,y). Even in this simply situation it is hard to visualize what is going on. But, you can contract one of the circles completely to a point geometrically (this you can visualize), but the other you cannnot. You can think of it as the smallest circle which will not contract to a point (i.e., it is in the center of the hole of the torus). We can flatten out the torus so that this part of the boundary protrudes. Then identify antipodal points of the circle. The effect of this is to remove the hole. We may identify the rest of the points. If you draw this or think about it a bit you will see that you get something homeomorphic to S^2. This is the exactly quotient space of the torus which is homeomorphic to a sphere. If you look back you will see that the operation was equivalent to mod-ing out by the wedge product of S^1 and S^1. Thus the smash product of any two pointed sets which are both homeomorphic to circles is the unique quotient space of the torus which is homeomorphic to a sphere.

I think, in the end, it is best to think of it as it relates to the wedge product.