This seems intuitive to me, since they admit a CW decomposition with finitely many cells. But I can't see how to prove it.
2026-04-30 08:05:54.1777536354
Do compact connected smooth manifolds admit the structure of a CW complex with a single 1-cell?
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This is false. Because if you see the cellular homology, then if $X$ has only one 1-cell, $H_1(X_1,X_0)$ would be a free group of $1$ genarator, but then if yoy compute $H_1(X)$ from cellular homology, then it has only one genarator. But there are many manifold with more than one genarator of $H_1$, for example torus.