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S³ x RP² and RP³ x S² are both smooth 5-manifolds with fundamental group Z/2 and universal cover S³ x S², so their homotopy groups are all the same. On the other hand, only the latter is orientable since RP³ is orientable but RP² isn't, so they have different values on H₅ and therefore can't be homotopy equivalent. (I think this example is in Hatcher somewhere.)