Mixing
Is there any general theory about existence of smooth structures of a topological manifold?
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I've read that there exists some topological manifold (of dimension $>3$) either with no smooth structure or having some multiple non-diffeomorphic smooth structures.
How to show that the topological manifold $mathbb{R}^4$ has no smooth structure?
And also what is about higher dimensions?
Is there any general theory about the existence of smooth structures of a topological manifold?
Thank You in advance.
geometry differential-geometry differential-topology smooth-manifolds
asked Jan 28 at 19:57
BijanDattaBijanDatta
309113
$endgroup$
add a comment |
$begingroup$
I've read that there exists some topological manifold (of dimension $>3$) either with no smooth structure or having some multiple non-diffeomorphic smooth structures.
How to show that the topological manifold $mathbb{R}^4$ has no smooth structure?
And also what is about higher dimensions?
Is there any general theory about the existence of smooth structures of a topological manifold?
Thank You in advance.
geometry differential-geometry differential-topology smooth-manifolds
asked Jan 28 at 19:57
BijanDattaBijanDatta
309113
$endgroup$
4
$begingroup$
$mathbb{R}^4$ has a smooth structure
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– Max
Jan 28 at 20:00
2
$begingroup$
See this question on Simon Donaldson and Michael Freedman in 1984 and the links given there. Try to search bit on this site here, there are many posts about this topic.
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– Dietrich Burde
Jan 28 at 20:02
2
$begingroup$
Yes, there is such a general theory. As a starting point you can look at Milnor's paper "Differential topology, 46 years later".
$endgroup$
– user98602
Jan 28 at 22:17
add a comment |
$begingroup$
I've read that there exists some topological manifold (of dimension $>3$) either with no smooth structure or having some multiple non-diffeomorphic smooth structures.
How to show that the topological manifold $mathbb{R}^4$ has no smooth structure?
And also what is about higher dimensions?
Is there any general theory about the existence of smooth structures of a topological manifold?
Thank You in advance.
geometry differential-geometry differential-topology smooth-manifolds
asked Jan 28 at 19:57
BijanDattaBijanDatta
309113
$endgroup$
I've read that there exists some topological manifold (of dimension $>3$) either with no smooth structure or having some multiple non-diffeomorphic smooth structures.
How to show that the topological manifold $mathbb{R}^4$ has no smooth structure?
And also what is about higher dimensions?
Is there any general theory about the existence of smooth structures of a topological manifold?
Thank You in advance.
geometry differential-geometry differential-topology smooth-manifolds
geometry differential-geometry differential-topology smooth-manifolds
asked Jan 28 at 19:57
BijanDattaBijanDatta
309113
asked Jan 28 at 19:57
BijanDattaBijanDatta
309113
asked Jan 28 at 19:57
BijanDattaBijanDatta
309113
asked Jan 28 at 19:57
BijanDattaBijanDatta
309113
asked Jan 28 at 19:57
BijanDattaBijanDatta
309113
309113
4
$begingroup$
$mathbb{R}^4$ has a smooth structure
$endgroup$
– Max
Jan 28 at 20:00
2
$begingroup$
See this question on Simon Donaldson and Michael Freedman in 1984 and the links given there. Try to search bit on this site here, there are many posts about this topic.
$endgroup$
– Dietrich Burde
Jan 28 at 20:02
2
$begingroup$
Yes, there is such a general theory. As a starting point you can look at Milnor's paper "Differential topology, 46 years later".
$endgroup$
– user98602
Jan 28 at 22:17
add a comment |
4
$begingroup$
$mathbb{R}^4$ has a smooth structure
$endgroup$
– Max
Jan 28 at 20:00
2
$begingroup$
See this question on Simon Donaldson and Michael Freedman in 1984 and the links given there. Try to search bit on this site here, there are many posts about this topic.
$endgroup$
– Dietrich Burde
Jan 28 at 20:02
2
$begingroup$
Yes, there is such a general theory. As a starting point you can look at Milnor's paper "Differential topology, 46 years later".
$endgroup$
– user98602
Jan 28 at 22:17
4
4
$begingroup$
$mathbb{R}^4$ has a smooth structure
$endgroup$
– Max
Jan 28 at 20:00
$begingroup$
$mathbb{R}^4$ has a smooth structure
$endgroup$
– Max
Jan 28 at 20:00
2
2
$begingroup$
See this question on Simon Donaldson and Michael Freedman in 1984 and the links given there. Try to search bit on this site here, there are many posts about this topic.
$endgroup$
– Dietrich Burde
Jan 28 at 20:02
$begingroup$
See this question on Simon Donaldson and Michael Freedman in 1984 and the links given there. Try to search bit on this site here, there are many posts about this topic.
$endgroup$
– Dietrich Burde
Jan 28 at 20:02
2
2
$begingroup$
Yes, there is such a general theory. As a starting point you can look at Milnor's paper "Differential topology, 46 years later".
$endgroup$
– user98602
Jan 28 at 22:17
$begingroup$
Yes, there is such a general theory. As a starting point you can look at Milnor's paper "Differential topology, 46 years later".
$endgroup$
– user98602
Jan 28 at 22:17
add a comment |
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4
$begingroup$
$mathbb{R}^4$ has a smooth structure
$endgroup$
– Max
Jan 28 at 20:00
2
$begingroup$
See this question on Simon Donaldson and Michael Freedman in 1984 and the links given there. Try to search bit on this site here, there are many posts about this topic.
$endgroup$
– Dietrich Burde
Jan 28 at 20:02
2
$begingroup$
Yes, there is such a general theory. As a starting point you can look at Milnor's paper "Differential topology, 46 years later".
$endgroup$
– user98602
Jan 28 at 22:17