Nonhomeomorphic CW-complexes that are stably homeomorphic

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.

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Yes.

See example 1.4 in the paper "Cancellation laws in topological products" by E. Santillán, Mor fismos, vol. 2, no. 2, 1998, pp. 67-74, and also the original example by R. H. Fox depicted on p.284 of his famous paper "On a problem of S. Ulam concerning Cartesian products", Fundamenta Mathematicae, vol. 34, no. 1, 1947, pp. 278-287. All these examples are finite CW complexes.

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