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Nikolay Konovalov
Nikolay Konovalov
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I have a very stupid question. Let $M$ be a closed smooth manifold. In particular cases the homotopy type of the diffeomorphism group $Diff(M)$ can be very pathological. For example, in the case of $M=T^{n-2}\times S^2$ we have that the group $\pi_i(Diff(M))$ has infinite rank for any $n\geq 5$ and $0\leq i\leq n-3$. But my question is about an opposite situation. Does someone know when the diffeomorphism group $Diff(M)$ has the homotopy type of aanother closed smooth manifold itself? Or a more simple question, when $Diff(M)$ is at least a Poincare space?

I have a very stupid question. Let $M$ be a closed smooth manifold. In particular cases the homotopy type of the diffeomorphism group $Diff(M)$ can be very pathological. For example, in the case of $M=T^{n-2}\times S^2$ we have that the group $\pi_i(Diff(M))$ has infinite rank for any $n\geq 5$ and $0\leq i\leq n-3$. But my question is about an opposite situation. Does someone know when the diffeomorphism group $Diff(M)$ has the homotopy type of a closed smooth manifold itself? Or a more simple question, when $Diff(M)$ is at least a Poincare space?

I have a very stupid question. Let $M$ be a closed smooth manifold. In particular cases the homotopy type of the diffeomorphism group $Diff(M)$ can be very pathological. For example, in the case of $M=T^{n-2}\times S^2$ we have that the group $\pi_i(Diff(M))$ has infinite rank for any $n\geq 5$ and $0\leq i\leq n-3$. But my question is about an opposite situation. Does someone know when the diffeomorphism group $Diff(M)$ has the homotopy type of another closed smooth manifold? Or a more simple question, when $Diff(M)$ is at least a Poincare space?

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Nikolay Konovalov
Nikolay Konovalov
  • 617
  • 4
  • 10

Manifolds whose diffeomorphism group has the homotopy type of a manifold itself

I have a very stupid question. Let $M$ be a closed smooth manifold. In particular cases the homotopy type of the diffeomorphism group $Diff(M)$ can be very pathological. For example, in the case of $M=T^{n-2}\times S^2$ we have that the group $\pi_i(Diff(M))$ has infinite rank for any $n\geq 5$ and $0\leq i\leq n-3$. But my question is about an opposite situation. Does someone know when the diffeomorphism group $Diff(M)$ has the homotopy type of a closed smooth manifold itself? Or a more simple question, when $Diff(M)$ is at least a Poincare space?