I want to find a topological space $X$ such that $X$ and $T^2 = S^1 \times S^1$ have the same of homology and distinct homotopy, but I have no idea how to find it. Can you help me? Thank you very much.
2026-03-28 10:16:47.1774693007
Same homology and distinct homotopy on torus
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Hint: What's the simplest possible way to build a space with the homology groups of $T^2$? How could you put together simple pieces to end up with a connected space $X$ with $H_2(X)\cong\mathbb{Z}$ and $H_1(X)\cong\mathbb{Z}^2$? (If you've learned a general method to construct a space with specified homology groups, you could see what that method gives you in this case.)