Suppose I have a manifold which has a CW structure with cells $e^0 \cup e^1 \cup e^2$, where $e^i$ represents an $i$-cell. If I took the direct product of this manifold with another manifold which has cell structure say $e^0 \cup e^2$, is it acceptable to say that I get a manifold with a CW structure with cells $(e^0 \cup e^2) \times (e^0 \cup e^1 \cup e^2)=e^0 \cup e^1 \cup e^2 \cup e^2 \cup e^3 \cup e^4$?
Is the resulting manifold at least homotopy equivalent to a manifold with such a cell decomposition?