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I was trying to come up with a visual intuition of the smash product of two topological spaces, and ended up understanding it as the result of the following process: 1. start with the topological product, 2. cut along the wedge product of the two spaces (in two steps if need be), 3. stretch the sections until you can finally 4. join the sections along their cuts.

Here are three examples of how I visualized $S^1\land S^1$, $S^1\land[0,1]$, and $[0,1]\land [0,1]$.

I then wondered if this process of "cut, stretch, join" applied to any topological quotient. Does it? Also, is $[0,1]\land [0,1]$ really a flower?

Mehmet Ates
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Note that in all your examples, you are taking the smash product of two 1-manifolds. If we consider, say, the smash product $\overline{D} \land [0,1]$, where $\overline{D} = \{ x^2 + y^2 \leq 1\}$ is the closed unit disk in $\mathbb{R}^2$, and if we choose the basepoint $x_0$ of $[0,1]$ not to be an endpoint, then first we would have to cut along $\overline{D} \times \{ x_0 \}$, which is a two-dimensional cut along a three-dimensional space. This seems fine - it's like cutting a cucumber in half.

But then next we would have to cut along $\{ y_0 \} \times [0,x_0) \subset \overline{D} \times [0,x_0)$, which is a one-dimensional cut along a three-dimensional space. This would be like cutting along a one-dimensional line in the longtitudal direction of (a half of) the cucumber - what does that even mean?

Generally, (in the case of manifolds), it seems like cutting an $n$-dimensional manifold is only possible along an $(n-1)$-dimensional submanifold, and even then only under suitable conditions. (Maybe someone knows a relevant theorem in this direction?)

It follows, by a dimensional argument, that only the product of two $1$-manifolds will result in cuts all of which are one dimension lower than the space along which you're cutting, and hence can be visualized nicely as in your picture.

(By the way, I agree that $[0,1]\land [0,1]$ is a flower - another way to see it is to gradually shift the intersection points of the dotted lines with the boundary of the square to the middle.)

Steven
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    I didn't realize I was assuming the cuts were always along 1-manifolds. Thanks for pointing this out, I really appreciate it! Also, for the example you gave, would the smash product just be two hemi-balls joined by a point on their circumference? – Mehmet Ates Jun 20 '23 at 15:59
  • @MehmetAtes I think it depends on the basepoint: if you choose the basepoint $y_0$ on the boundary of the disk, I think this is indeed what happens, and this is just homeomorphic to the wedge sum of two closed $3$-disks (with basepoints on their bdry). If the $y_0$ is in the interior, you get the wedge product of two semi-balls with the bottom 'pushed inside', which I don't think is homeomorphic to the previous case. – Steven Jun 21 '23 at 07:33