Suppose there are compact sets $K_1,K_2\subset\mathbb{R}^n$ such that $K_1\times \mathbb{R}\cong K_2\times \mathbb{R}$, but $K_1\ncong K_2$.

What is the minimum of $n$?

Comments

• The spherical suspension over the Poincaré homology sphere and $\mathbb{S}^4$ provide an example for $n=5$.

• If instead of $\mathbb{R}$ one takes the closed interval $[0,1]$, then examples of planar compact sets are shown. See also this question

  

• If instead of $\mathbb{R}$ one takes the closed interval $[0,1]$, then examples of plane compact sets are shown. See also this question.

• If one removes the assumption of compactness, then examples can be constructed in a similar way (for $n=2$).

Suppose there are compact sets $K_1,K_2\subset\mathbb{R}^n$ such that $K_1\times \mathbb{R}\cong K_2\times \mathbb{R}$, but $K_1\ncong K_2$.

What is the minimum of $n$?

Comments

• The spherical suspension over the Poincaré homology sphere and $\mathbb{S}^4$ provide an example for $n=5$.

• If instead of $\mathbb{R}$ one takes the closed interval $[0,1]$, then examples of planar compact sets are shown below. See also this question.

  

• If one removes the assumption of compactness, then examples can be constructed in a similar way (for $n=2$).