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revised How to show $[\omega]=0$ implies $[\omega^n]=0$?
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comment Example of a pair of non-cobordant manifolds
The cone on $\Bbb C P^2$ is not a topological manifold. In fact, the cone on any space $X$ that does not have the same integral homology as a sphere cannot possibly be a manifold, because for such spaces we have $\tilde{H}_\ast(X, X - \text{cone pt}; \Bbb Z) \not \cong \Bbb Z$.
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