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comment characteristic classes of homotopy equivalent manifolds
@DannyRuberman Thanks for the correction, I tried to answer the question too fast. The $\lambda$-invariant for Milnor's exotic spheres are defined in terms of Pontryagin classes, but of course are not related to the (trivial) Pontryagin classes of the exotic sphere itself but instead those of an $8$-manifold it bounds.
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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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