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I'm reading "An Introduction to Manifolds" by Tu. In this book, the definition of a topological manifold is a Hausdorff, second countable locally Euclidean space and the definition of a smooth manifold is a topological manifold with a maximal (pairwise $C^\infty$-compatible) atlas.

I know that given a topological manifold $M$ and an atlas $\mathfrak{A}$ on $M$, there is a unique maximal atlas on $M$ that contains $\mathfrak{A}$. My question is that is it possible to have two distinct maximal atlases on the topological manifold $M$ (that is,they are not required to contain a given atlas on $M$)?

Alex Ortiz
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baby bunny
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  • If I understand correctly, then Proposition 5.10 of Tu may be understood in the following way as illustrated in the answer by Alex Ortiz. If $A_1$ and $A_2$ are two atlases which are not compatible with each other, then they are contained in two different maximal atlases $\mathcal{M}_1$ and $\mathcal{M}_2$, where $\mathcal{M}_1 \neq \mathcal{M}_2$. – rainman Jun 10 '20 at 16:43

2 Answers2

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In general there is not a unique maximal atlas on the manifold $M$. Consider $\Bbb R$ and the charts $(\Bbb R,\operatorname{Id})$ and $(\Bbb R,f)$, where $f(x) = x^3$. Each of these charts cover $\Bbb R$, but it's easy to check they are not smoothly compatible, so they generate distinct maximal atlases. However, the smooth manifolds we get by equipping $\mathbb R$ with each of these maximal atlases are diffeomorphic.

Exotic spheres are a famous example of spaces having multiple smooth structures that are not diffeomorphic to one another.

Alex Ortiz
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The checked answer misses the point of the question.

Typically, we do not discuss atlases in the context of topological manifolds. That is more of a social convention, because we can certainly introduce the notion of topological atlas.

A topological atlas of a topological space $M$ is a collection $\{ \phi_i : U_i \rightarrow \mathbb R^n \}$ where each $U_i$ is an open subset of $M$ and $\phi_i$ is a homeomorphism, and the collection $U_i$ covers $M$.

With the above definition, any union of topological atlases is already a topological atlas, and so there exists a unique maximal topological atlas for each topological manifold. More specifically, we use Zorn's lemma to conclude that every atlas is contained in a maximal atlas, and then we see that all maximal atlases are, in fact, one and the same atlas.

If we were working in the smooth category (or any other more complicated category), then we would need to define the atlas via requirements on the transition functions. However, that is not necessary here: we have got homeomorphisms which are the transition functions $$ \phi_j \phi_i^{-1}: \phi_i( U_i \cap U_j ) \rightarrow \phi_j( U_i \cap U_j ). $$ This is the usual compatibility condition of the coordinate charts, and it holds automatically for any collection of topological coordinate charts.

shuhalo
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