Can one find two finite CW-complexes $X$ and $Y$ such that $X \times I$ is homeomorphic to $Y \times I$, where $I = [0, 1]$, but $X$ is not homeomorphic to $Y$? I know how to find such topological spaces $X$ and $Y$, but not CW-complexes.
Nonhomeomorphic CW-complexes that are stably homeomorphic
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geometric-topology