For curves this follows from the classification of (2-dimensional topological) surfaces, and for simply-connected surfaces this follows from [Freedman's theorem.][1] 

My former colleagues Anatoly Libgober and John Wood found examples of pairs of 3-folds which are complete intersections and are homotopy equivalent but not diffeomorphic, in fact have distinct Pontryagin classes. See [Example 9.2][2]. Since in this case $H^4(M;\mathbb{Z})\cong \mathbb{Z}$, this implies that the manifolds are not homeomorphic by the [topological invariance of rational Pontryagin classes][3] (see Ben Wieland's comment). 


  [1]: https://mathworld.wolfram.com/FreedmanTheorem.html
  [2]: https://doi.org/10.1016/0040-9383(82)90024-6
  [3]: https://encyclopediaofmath.org/wiki/Pontryagin_class#Topological_invariance.