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Nohomeomorphic Nonhomeomorphic CW-complexes that are "stably" homeomorphic
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Nohomeomorphic CW-complexes that are "stably" homeomorphic

Do there exist CW-complexes $X$ and $Y$ that are not homeomorphic, but $X \times I$ and $Y \times I$ are homeomorphic? Here $I$ denotes the unit interval $[0, 1]$.