All compatible charts are included in the same atlas, forming a maximal atlas.
But can there be more than one maximal(incompatible) atlas for the same topological manifold?
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It's not really clear what you are asking. Is your question: "Can we define more than one (not compatible) maximal atlas on the same topological manifold"? – Ennar Nov 19 '18 at 10:14
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@Ennar I think that's what I wanted to say. Sorry, I'm self-learning this... ;) – An old man in the sea. Nov 19 '18 at 10:21
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I found the answer here, I think. @Ennar Is my comment to the answer in the link correct? https://math.stackexchange.com/questions/852386/confused-about-the-possibility-of-different-differentiable-structures?rq=1 – An old man in the sea. Nov 19 '18 at 10:31
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Yes. Any atlas consists only of charts that are compatible. Take any incompatible charts and they need to belong to different atlases. However, in the example there, you will get two incompatible maximal atlases, but they still give diffeomorphic manifolds. Take a look here. – Ennar Nov 19 '18 at 10:41
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@Ennar many thanks ! ;) – An old man in the sea. Nov 19 '18 at 10:45