Let $X=S^1 \times I$ be a cylinder, where $S^1$ is the 1-dimensional circle. If we glue the "bottom" boundary $S^1 \times 0$ and the "top" boundary $S^1\times 1$ by a homeomorphism sending $x\times 0$ to $x\times 1$, the resulting manifold is homeomorphic to a torus.
If we chose another homeomorphism to glue boundaries, can we get a manifold that is not homeomorphic to a torus? (I only consider oriented manifolds so Klein bottle are omitted.)
Or no matter what homeomorphism we choose, is the resulting manifold is homeomorphic to a torus?