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2021 Alibaba Global Math Competition, 11
Let $M$ be a compact orientable $2n$-manifold with boundary, where $n \ge 2$. Suppose that $H_0(M;\mathbb{Q}) \cong \mathbb{Q}$ and $H_i(M;\mathbb{Q})=0$ for $i>0$. Prove that the order of $H_{n-1}(\partial M; \mathbb{Z})$ is a square number.
manifold topology boundary algebraic topology homology