I am not a topologist and hence don't know how to make this answer rigorous; but, intuitively speaking, it's obvious that any "small" simplex is homotopic to any other "small" simplex on a connected $n$-manifold (translate it using the atlases, then resize it). Hence the rank of $H_n(X)$ for connected $n$-manifolds $X$ is clearly at most 1 (any simplex is a multiple of any other simplex).
A manifold $X$ being non-orientable means that any $n$-cycle (simplex) $S$ is homotopic (and in fact equal, modulo $n$-boundaries) to $-S$. Hence all elements of $H_n(X)$ have order at most 2, hence the Betti number is 0 (because the Betti number only counts the number of *infinite* dimensions). $\blacksquare$
clemens
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2026-05-07T17:21:04Z (4 months ago)