I am looking for $X$ which has the same integral homology and fundamental group as the torus, $T$, which is not homotopic equivalent to $T$.
At first, I considered $S^1\vee S^1 \vee S^2$, but $\pi_1(S^1\vee S^1 \vee S^2)$ is not $\mathbb Z^2$, so this example doesn't work.
This problem came up in a past qual exam, and so I believe there is a "simple" example, where proving that $X$ is not homotopic equivalent to $T$ is not too "difficult". The argument would probably involve showing that they have distinct cohomology rings.
I would appreciate any help/hint.