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edit approved Apr 24, 2017 at 17:58
Dario
edit approved Apr 24, 2017 at 17:58
Dario
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Check the paper Antonelli, P. L.; Burghelea, D.; Kahn, P. J. The non-finite homotopy type of some diffeomorphism groups.The non-finite homotopy type of some diffeomorphism groups. Topology 11 (1972), 1–49.

There they mention an old result of Palais (Homotopy theory of infinite dimensional manifoldsHomotopy theory of infinite dimensional manifolds. Topology 5 (1966), 1-16) which states that the identity component of ${\rm Diff}_0(M)$ has the homotopy type of a countable $CW$-complex.

In their paper, Antonelli, Burghelea and Kahn prove that for many smooth manifolds (including spheres of dimension $\geq 7$) the group ${\rm Diff}_0(M)$ does not have the homotopy type of a finite $CW$-complex. (This is highly nontrivial.)

Above, by diffeomorphisms they mean smooth diffeomorphisms and the topology is the $C^\infty$-topology.

Check the paper Antonelli, P. L.; Burghelea, D.; Kahn, P. J. The non-finite homotopy type of some diffeomorphism groups. Topology 11 (1972), 1–49.

There they mention an old result of Palais (Homotopy theory of infinite dimensional manifolds. Topology 5 (1966), 1-16) which states that the identity component of ${\rm Diff}_0(M)$ has the homotopy type of a countable $CW$-complex.

In their paper, Antonelli, Burghelea and Kahn prove that for many smooth manifolds (including spheres of dimension $\geq 7$) the group ${\rm Diff}_0(M)$ does not have the homotopy type of a finite $CW$-complex. (This is highly nontrivial.)

Above, by diffeomorphisms they mean smooth diffeomorphisms and the topology is the $C^\infty$-topology.

Check the paper Antonelli, P. L.; Burghelea, D.; Kahn, P. J. The non-finite homotopy type of some diffeomorphism groups. Topology 11 (1972), 1–49.

There they mention an old result of Palais (Homotopy theory of infinite dimensional manifolds. Topology 5 (1966), 1-16) which states that the identity component of ${\rm Diff}_0(M)$ has the homotopy type of a countable $CW$-complex.

In their paper, Antonelli, Burghelea and Kahn prove that for many smooth manifolds (including spheres of dimension $\geq 7$) the group ${\rm Diff}_0(M)$ does not have the homotopy type of a finite $CW$-complex. (This is highly nontrivial.)

Above, by diffeomorphisms they mean smooth diffeomorphisms and the topology is the $C^\infty$-topology.

corrected typo
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Ricardo Andrade
Ricardo Andrade
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Check the paper Antonelli, P. L.; Burghelea, D.; Kahn, P. J. The non-finite homotopy type of some diffeomorphism groups. Topology 11 (1972), 1–49.

There theythey mention an old result of Palais (HomotopvHomotopy theory of infinite dimensional manifolds. Topology 5 (1966), 1-16) which states that the identity component of  ${\rm Diff}_0(M)$ has the homotopy type of a countable $CW$-complex.

In their paper, Antonelli,Burghelea Burghelea and Kahn proveKahn prove that for many smooth manifolds (including spheres of dimension $\geq 7$) thethe group ${\rm Diff}_0(M)$ does not have the homotopy type of a finite $CW$-complex. (This is highly nontrivial.)

Above, by diffeomorphisms theythey mean smooth diffeomorphismdiffeomorphisms and the topology is the $C^\infty$-topology.

Check the paper Antonelli, P. L.; Burghelea, D.; Kahn, P. J. The non-finite homotopy type of some diffeomorphism groups. Topology 11 (1972), 1–49.

There they mention an old result of Palais (Homotopv theory of infinite dimensional manifolds. Topology 5 (1966), 1-16) which states that the identity component of  ${\rm Diff}_0(M)$ has the homotopy type of a countable $CW$-complex.

In their paper, Antonelli,Burghelea and Kahn prove that for many smooth manifolds (including spheres of dimension $\geq 7$) the group ${\rm Diff}_0(M)$ does not have the homotopy type of a finite $CW$-complex. (This is highly nontrivial.)

Above, by diffeomorphisms they mean smooth diffeomorphism and the topology is the $C^\infty$-topology.

Check the paper Antonelli, P. L.; Burghelea, D.; Kahn, P. J. The non-finite homotopy type of some diffeomorphism groups. Topology 11 (1972), 1–49.

There they mention an old result of Palais (Homotopy theory of infinite dimensional manifolds. Topology 5 (1966), 1-16) which states that the identity component of ${\rm Diff}_0(M)$ has the homotopy type of a countable $CW$-complex.

In their paper, Antonelli, Burghelea and Kahn prove that for many smooth manifolds (including spheres of dimension $\geq 7$) the group ${\rm Diff}_0(M)$ does not have the homotopy type of a finite $CW$-complex. (This is highly nontrivial.)

Above, by diffeomorphisms they mean smooth diffeomorphisms and the topology is the $C^\infty$-topology.

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Liviu Nicolaescu
Liviu Nicolaescu
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Check the paper Antonelli, P. L.; Burghelea, D.; Kahn, P. J. The non-finite homotopy type of some diffeomorphism groups. Topology 11 (1972), 1–49.

There they mention an old result of Palais (Homotopv theory of infinite dimensional manifolds. Topology 5 (1966), 1-16) which states that the identity component of ${\rm Diff}_0(M)$ has the homotopy type of a countable $CW$-complex.

In their paper, Antonelli,Burghelea and Kahn prove that for many smooth manifolds (including spheres of dimension $\geq 7$) the group ${\rm Diff}_0(M)$ does not have the homotopy type of a finite $CW$-complex. (This is highly nontrivial.)

Above, by diffeomorphisms they mean smooth diffeomorphism and the topology is the $C^\infty$-topology.