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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.

Bob
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  • A smooth map is necessarily continuous, so a smooth map with smooth inverse is a continuous map with continuous inverse. Diffeomorphisms are homeomorphisms, too. –  Sep 21 '16 at 17:19
  • If you're asking whether a set can have two different $C^r$ compatible atlases that induce different topologies, then the answer is certainly yes. Since every (nonempty) positive-dimensional manifold has the cardinality of $\mathbb R$, we can define one $C^r$-compatible atlas on $\mathbb R$ that induces the usual topology on $\mathbb R$, and another that induces the topology of, say, $S^2$. – Jack Lee Sep 21 '16 at 17:48

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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.

John Hughes
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  • The question appears to be asking the converse. –  Sep 21 '16 at 17:19
  • Now that you mention it, it does seem to be asking that, and your answer and Jack Lee's comment both answer that nicely. On the other hand, the OP seems to have selected my answer, so perhaps my misunderstanding matched OP's mis-statement or something. :) I'm leaving it here because someone else might arrive here when trying to ask a similar question. – John Hughes Sep 21 '16 at 17:58
  • Certainly I've no objection to more people learning about Milnor's beautiful paper :) –  Sep 21 '16 at 18:02