If two atlases have the same differential structure (they're both $C^r$) do they necessarily have the same topology? My thought is, since an atlas induces both the differentiable structure and the topology then the answer to my question might be yes.
2026-04-07 02:00:37.1775527237
Differentiable Structure of Atlas and its Relationship to Topology
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No, because there are smooth manifolds (i.e., both $C^{\infty}$) that are homoemorphic but not diffeomorphic. The classic reference for this is Milnor's paper on differentiable structures on the 7-sphere, ca 1960.