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Kim Morrison
Kim Morrison
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Let $M$ be a (real) manifold. Recall that an analytic structure on $M$ is an atlas such that all transition maps are real-analytic (and maximal with respect to this property). (There's also a sheafy definition.) So in particular being analytic is a structure, not a property.

Q1: Is it true that any topological manifold can be equipped with an analytic structure? Can

Q2: Can any $C^\infty$ manifold (replace "analytic" by "$C^\infty$" in the previousfirst paragraph) be equipped with an analytic structure (consistent with the smooth structure)?

Let $M$ be a (real) manifold. Recall that an analytic structure on $M$ is an atlas such that all transition maps are real-analytic (and maximal with respect to this property). (There's also a sheafy definition.) So in particular being analytic is a structure, not a property.

Is it true that any topological manifold can be equipped with an analytic structure? Can any $C^\infty$ manifold (replace "analytic" by "$C^\infty$" in the previous paragraph) be equipped with an analytic structure (consistent with the smooth structure)?

Let $M$ be a (real) manifold. Recall that an analytic structure on $M$ is an atlas such that all transition maps are real-analytic (and maximal with respect to this property). (There's also a sheafy definition.) So in particular being analytic is a structure, not a property.

Q1: Is it true that any topological manifold can be equipped with an analytic structure?

Q2: Can any $C^\infty$ manifold (replace "analytic" by "$C^\infty$" in the first paragraph) be equipped with an analytic structure (consistent with the smooth structure)?

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Theo Johnson-Freyd
Theo Johnson-Freyd
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Can every manifold be given an analytic structure?

Let $M$ be a (real) manifold. Recall that an analytic structure on $M$ is an atlas such that all transition maps are real-analytic (and maximal with respect to this property). (There's also a sheafy definition.) So in particular being analytic is a structure, not a property.

Is it true that any topological manifold can be equipped with an analytic structure? Can any $C^\infty$ manifold (replace "analytic" by "$C^\infty$" in the previous paragraph) be equipped with an analytic structure (consistent with the smooth structure)?