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 $[0, 1]$.
Yes. Take $X$ a punctured torus ($T^2\setminus$open disk) and $Y$ a three-punctured $S^2$. Then $X\times I=Y\times I$ is a genus 2 handlebody.