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symplectomorphic
symplectomorphic
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It's a result of low-dimensional topology that in dimensions 3 and lower, two manifolds are homeomorphic if and only if they are diffeomorphic. The 28 differentiable structures on Milnor's 7-spherespheres give a nice counterexamplecounterexamples to this result in dimension 7, and exotic $\mathbb{R}^4$ gives a's give nice counterexamplecounterexamples in dimension 4. But I don't know about dimensions 5 and 6. Is the result true or false in dimensions 5 and 6? And, and if false, what are some classic counterexamples, and do stronger constraints -- say compactness or closedness -- happen to make it true?

It's a result of low-dimensional topology that in dimensions 3 and lower, two manifolds are homeomorphic if and only if they are diffeomorphic. The 28 differentiable structures on Milnor's 7-sphere give a nice counterexample to this result in dimension 7, and exotic $\mathbb{R}^4$ gives a nice counterexample in dimension 4. But I don't know about dimensions 5 and 6. Is the result true or false in dimensions 5 and 6, and if false, what are some classic counterexamples?

It's a result of low-dimensional topology that in dimensions 3 and lower, two manifolds are homeomorphic if and only if they are diffeomorphic. Milnor's 7-spheres give nice counterexamples to this result in dimension 7, and exotic $\mathbb{R}^4$'s give nice counterexamples in dimension 4. But I don't know about dimensions 5 and 6. Is the result true or false in dimensions 5 and 6? And, if false, what are some classic counterexamples, and do stronger constraints -- say compactness or closedness -- happen to make it true?

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symplectomorphic
symplectomorphic
  • 1.2k
  • 11
  • 19

It's a result of low-dimensional topology that in dimensions 3 and lower, two manifolds are homeomorphic if and only if they are diffeomorphic. The 28 differentiable structures on Milnor's 7-sphere give a nice counterexample to this result in dimension 7, and I have heard the result is also falseexotic $\mathbb{R}^4$ gives a nice counterexample in dimension 4. But I don't know about dimensions 5 and 6. So my question comes in two parts:

-- is there a relatively famous or classic counterexample to the result in dimension 4?

-- isIs the result true or false in dimensions 5 and 6, and if false, what are some classic counterexamples?

It's a result of low-dimensional topology that in dimensions 3 and lower, two manifolds are homeomorphic if and only if they are diffeomorphic. The 28 differentiable structures on Milnor's 7-sphere give a nice counterexample to this result in dimension 7, and I have heard the result is also false in dimension 4. But I don't know about dimensions 5 and 6. So my question comes in two parts:

-- is there a relatively famous or classic counterexample to the result in dimension 4?

-- is the result true or false in dimensions 5 and 6, and if false, what are some classic counterexamples?

It's a result of low-dimensional topology that in dimensions 3 and lower, two manifolds are homeomorphic if and only if they are diffeomorphic. The 28 differentiable structures on Milnor's 7-sphere give a nice counterexample to this result in dimension 7, and exotic $\mathbb{R}^4$ gives a nice counterexample in dimension 4. But I don't know about dimensions 5 and 6. Is the result true or false in dimensions 5 and 6, and if false, what are some classic counterexamples?

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symplectomorphic
symplectomorphic
  • 1.2k
  • 11
  • 19

exotic differentiable structures on manifolds in dimensions 5 and 6

It's a result of low-dimensional topology that in dimensions 3 and lower, two manifolds are homeomorphic if and only if they are diffeomorphic. The 28 differentiable structures on Milnor's 7-sphere give a nice counterexample to this result in dimension 7, and I have heard the result is also false in dimension 4. But I don't know about dimensions 5 and 6. So my question comes in two parts:

-- is there a relatively famous or classic counterexample to the result in dimension 4?

-- is the result true or false in dimensions 5 and 6, and if false, what are some classic counterexamples?