The Top non-zero Betti number of a closed oriented manifold is one. is it true for the general manifold or not?
2026-02-22 22:38:12.1771799892
Top non-zero Betti number of connected manifold of finite type.
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No it isn't true in general. For example, take the nonorientable surface $(\mathbb{RP}^2)^{\# n}$ for $n>2$. The only nontrivial reduced homology group is $\tilde{H}_1=\mathbb{Z}^{n-1}\oplus\mathbb{Z}/2$ which doesn't have rank $1$.