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Example of “practical” applications of Donaldson Invariants
The 2019 Stack Overflow Developer Survey Results Are Inholomorphic exoticnessIs $T^1 S^5$ abstractly diffeomorphic to $S^4times S^5$?$D^mcup_h D^m$, joining $D^m amalg D^m$ along the boundary $partial D^m$Help understanding manifolds and topological spacesResearching for differential invariantsComputing geodesic distances from structural dataAn example on singular homologyHow are topological invariants obtained from TQFTs used in practice?How to think about exotic differentiable structures in manifolds?Smoothing non-differentiable manifolds?
$begingroup$
I'm studying Donaldson Invariants from chapter 9 of The wild world of 4-manifolds by Scorpan, and I'm looking for an example where they're used to distinguish two 4-manifolds which are homeomorphic but not diffeomorphic.
I know that these invariants are quite hard to compute, and honestly I don't even know where to start in order to find such manifolds; are there any "famous examples" well-known in the literature?
Thank you.
algebraic-topology differential-topology 4-manifolds topological-quantum-field-theory
edited Apr 4 at 11:25
Andrews
1,2812423
asked Mar 30 at 16:34
Giuseppe BargagnatiGiuseppe Bargagnati
1,251514
$endgroup$
add a comment |
$begingroup$
I'm studying Donaldson Invariants from chapter 9 of The wild world of 4-manifolds by Scorpan, and I'm looking for an example where they're used to distinguish two 4-manifolds which are homeomorphic but not diffeomorphic.
I know that these invariants are quite hard to compute, and honestly I don't even know where to start in order to find such manifolds; are there any "famous examples" well-known in the literature?
Thank you.
algebraic-topology differential-topology 4-manifolds topological-quantum-field-theory
edited Apr 4 at 11:25
Andrews
1,2812423
asked Mar 30 at 16:34
Giuseppe BargagnatiGiuseppe Bargagnati
1,251514
$endgroup$
$begingroup$
'are there any "famous examples" well-known in the literature?' - examples of 4-manifold pairs that are homeomorphic but not diffeomorphic? If the answer is yes, you can cook up very easy examples. I think what you're interested in are 4-manifold pairs that are both smooth but have different smooth structure. Please correct me if I'm mistaken.
$endgroup$
– Prototank
Mar 30 at 21:36
$begingroup$
Yes, I'm interested in smooth 4-manifolds that do not admit smooth structures such that they are diffeomorphic, but that are homeomorphic; in the question when I write manifold I always mean "smooth" manifold.
$endgroup$
– Giuseppe Bargagnati
Mar 30 at 21:53
1
$begingroup$
See arxiv.org/pdf/0812.1883.pdf
$endgroup$
– Nick L
Apr 4 at 12:24
add a comment |
$begingroup$
I'm studying Donaldson Invariants from chapter 9 of The wild world of 4-manifolds by Scorpan, and I'm looking for an example where they're used to distinguish two 4-manifolds which are homeomorphic but not diffeomorphic.
I know that these invariants are quite hard to compute, and honestly I don't even know where to start in order to find such manifolds; are there any "famous examples" well-known in the literature?
Thank you.
algebraic-topology differential-topology 4-manifolds topological-quantum-field-theory
edited Apr 4 at 11:25
Andrews
1,2812423
asked Mar 30 at 16:34
Giuseppe BargagnatiGiuseppe Bargagnati
1,251514
$endgroup$
I'm studying Donaldson Invariants from chapter 9 of The wild world of 4-manifolds by Scorpan, and I'm looking for an example where they're used to distinguish two 4-manifolds which are homeomorphic but not diffeomorphic.
I know that these invariants are quite hard to compute, and honestly I don't even know where to start in order to find such manifolds; are there any "famous examples" well-known in the literature?
Thank you.
algebraic-topology differential-topology 4-manifolds topological-quantum-field-theory
algebraic-topology differential-topology 4-manifolds topological-quantum-field-theory
edited Apr 4 at 11:25
Andrews
1,2812423
asked Mar 30 at 16:34
Giuseppe BargagnatiGiuseppe Bargagnati
1,251514
edited Apr 4 at 11:25
Andrews
1,2812423
asked Mar 30 at 16:34
Giuseppe BargagnatiGiuseppe Bargagnati
1,251514
edited Apr 4 at 11:25
Andrews
1,2812423
edited Apr 4 at 11:25
Andrews
1,2812423
edited Apr 4 at 11:25
Andrews
1,2812423
1,2812423
asked Mar 30 at 16:34
Giuseppe BargagnatiGiuseppe Bargagnati
1,251514
asked Mar 30 at 16:34
Giuseppe BargagnatiGiuseppe Bargagnati
1,251514
asked Mar 30 at 16:34
Giuseppe BargagnatiGiuseppe Bargagnati
1,251514
1,251514
$begingroup$
'are there any "famous examples" well-known in the literature?' - examples of 4-manifold pairs that are homeomorphic but not diffeomorphic? If the answer is yes, you can cook up very easy examples. I think what you're interested in are 4-manifold pairs that are both smooth but have different smooth structure. Please correct me if I'm mistaken.
$endgroup$
– Prototank
Mar 30 at 21:36
$begingroup$
Yes, I'm interested in smooth 4-manifolds that do not admit smooth structures such that they are diffeomorphic, but that are homeomorphic; in the question when I write manifold I always mean "smooth" manifold.
$endgroup$
– Giuseppe Bargagnati
Mar 30 at 21:53
1
$begingroup$
See arxiv.org/pdf/0812.1883.pdf
$endgroup$
– Nick L
Apr 4 at 12:24
add a comment |
$begingroup$
'are there any "famous examples" well-known in the literature?' - examples of 4-manifold pairs that are homeomorphic but not diffeomorphic? If the answer is yes, you can cook up very easy examples. I think what you're interested in are 4-manifold pairs that are both smooth but have different smooth structure. Please correct me if I'm mistaken.
$endgroup$
– Prototank
Mar 30 at 21:36
$begingroup$
Yes, I'm interested in smooth 4-manifolds that do not admit smooth structures such that they are diffeomorphic, but that are homeomorphic; in the question when I write manifold I always mean "smooth" manifold.
$endgroup$
– Giuseppe Bargagnati
Mar 30 at 21:53
1
$begingroup$
See arxiv.org/pdf/0812.1883.pdf
$endgroup$
– Nick L
Apr 4 at 12:24
$begingroup$
'are there any "famous examples" well-known in the literature?' - examples of 4-manifold pairs that are homeomorphic but not diffeomorphic? If the answer is yes, you can cook up very easy examples. I think what you're interested in are 4-manifold pairs that are both smooth but have different smooth structure. Please correct me if I'm mistaken.
$endgroup$
– Prototank
Mar 30 at 21:36
$begingroup$
'are there any "famous examples" well-known in the literature?' - examples of 4-manifold pairs that are homeomorphic but not diffeomorphic? If the answer is yes, you can cook up very easy examples. I think what you're interested in are 4-manifold pairs that are both smooth but have different smooth structure. Please correct me if I'm mistaken.
$endgroup$
– Prototank
Mar 30 at 21:36
$begingroup$
Yes, I'm interested in smooth 4-manifolds that do not admit smooth structures such that they are diffeomorphic, but that are homeomorphic; in the question when I write manifold I always mean "smooth" manifold.
$endgroup$
– Giuseppe Bargagnati
Mar 30 at 21:53
$begingroup$
Yes, I'm interested in smooth 4-manifolds that do not admit smooth structures such that they are diffeomorphic, but that are homeomorphic; in the question when I write manifold I always mean "smooth" manifold.
$endgroup$
– Giuseppe Bargagnati
Mar 30 at 21:53
1
1
$begingroup$
See arxiv.org/pdf/0812.1883.pdf
$endgroup$
– Nick L
Apr 4 at 12:24
$begingroup$
See arxiv.org/pdf/0812.1883.pdf
$endgroup$
– Nick L
Apr 4 at 12:24
add a comment |
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$begingroup$
'are there any "famous examples" well-known in the literature?' - examples of 4-manifold pairs that are homeomorphic but not diffeomorphic? If the answer is yes, you can cook up very easy examples. I think what you're interested in are 4-manifold pairs that are both smooth but have different smooth structure. Please correct me if I'm mistaken.
$endgroup$
– Prototank
Mar 30 at 21:36
$begingroup$
Yes, I'm interested in smooth 4-manifolds that do not admit smooth structures such that they are diffeomorphic, but that are homeomorphic; in the question when I write manifold I always mean "smooth" manifold.
$endgroup$
– Giuseppe Bargagnati
Mar 30 at 21:53
1
$begingroup$
See arxiv.org/pdf/0812.1883.pdf
$endgroup$
– Nick L
Apr 4 at 12:24