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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeOct 30th 2014

started a section on the homotopy type of the diffeomorphism group and recorded the case for closed orientable surfaces

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeOct 31st 2014

added a few more facts under Homotopy type and mapping class group, all taken from Hatcher’s review (pdf), also dug out some of the references (unfortunately no citation detail in that review).

Is there a more recent/more comprehensive such review? Textbook account? (Unfortunatley Hatcher’s review is not dated.)

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeOct 31st 2014

There’s 2012 in the URL.

And on his homepage it says

A talk at the 50th Cornell Topology Festival in May 2012 sketching some highlights of what’s known about the homotopy types of diffeomorphism groups of smooth manifolds. A full history would of course be impossible in an hour talk.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeOct 31st 2014

Ah. Should have seen that. Thanks! Am adding it now…

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeOct 28th 2021

I have added the following pointers, claiming dis-proof of the analogue of the Smale conjecture in 4d:

• Tadayuki Watanabe, Some exotic nontrivial elements of the rational homotopy groups of $Diff(S^4)$ (arXiv:1812.02448)

• Tadayuki Watanabe, Addendum to: Some exotic nontrivial elements of the rational homotopy groups of $Diff(S^4)$ (homological interpretation) (arXiv:2109.01609)