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nLab > Latest Changes: diffeomorphism group

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  1.  
    • 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(S4) (arXiv:1812.02448)

    • Tadayuki Watanabe, Addendum to: Some exotic nontrivial elements of the rational homotopy groups of Diff(S4) (homological interpretation) (arXiv:2109.01609)

    diff, v10, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 24th 2023

    added pointer to today’s

    • J. S. Dowker, Note on the structure constants for the diffeomorphisms of the two-sphere [arXiv:2301.09487]

    diff, v12, current

    • CommentRowNumber7.
    • CommentAuthorperezl.alonso
    • CommentTimeFeb 19th 2025

    Was there somewhere in the nLab a discussion of the stack X//Diff(X) (and maybe its geometric realization given the results listed in this page for 2d surfaces), or am I imagining this?

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeFeb 19th 2025

    I was recently talking about π1Maps(Σ,S2)⫽π0Diff(Σ), for Σ a surface. Maybe that’s what you saw.

    Parts of this discussion is at 2-Cohomotopy moduli of surfaces, more discussion is on pp. 21 in Engineering of Anyons on M5-Probes.

    • CommentRowNumber9.
    • CommentAuthorperezl.alonso
    • CommentTimeFeb 19th 2025

    Hmm, I don’t recall this, actually. What I had in mind was something more along the lines of Section 5.3 in this paper by Hořava. But it seems the second thing you mentioned is related?

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeFeb 20th 2025

    Alternatively, the expression Σ⫽Diff(Σ) plays a central role in the discussion general covariance – in HoTT.

    • CommentRowNumber11.
    • CommentAuthorperezl.alonso
    • CommentTimeFeb 20th 2025

    Yes, I think that is more along the lines of what I remembered, thanks.

    • CommentRowNumber12.
    • CommentAuthorperezl.alonso
    • CommentTimeFeb 25th 2025
    • (edited Feb 25th 2025)

    Are there any concrete results concerning diffeomorphism (higher) groups of orbifolds? Say a concrete presentation for 2d orbisurfaces or similar?

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeFeb 25th 2025

    The mapping class group π0Diff(Σ) (the connected component group of the group of diffeomorphisms) for orbi-surfaces Σ was recently discussed in:

    • Jonas Flechsig: Braid groups and mapping class groups for 2-orbifolds, PhD thesis Bielefeld (2023) [doi:10.4119/unibi/2979933]

    • Jonas Flechsig: Mapping class groups for 2-orbifolds [arXiv:2305.04272]

    • Jonas Flechsig: Braid groups and mapping class groups for 2-orbifolds [arXiv:2305.04273]

    • CommentRowNumber14.
    • CommentAuthorperezl.alonso
    • CommentTimeFeb 25th 2025

    Re#13 thanks, added a line to that effect in the references section

    diff, v13, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeMar 9th 2025
    • (edited Mar 9th 2025)

    Added the statement (here) that the connected components of diffeomorphism groups of compact smooth surfaces with boundary, fixing the boundary pointswise, is contractible.

    Together with pointer to:

    • C. J. Earle, A. Schatz: Teichmüller theory for surfaces with boundary, J. Differential Geom. 4 2 (1970) 169-185 [doi:10.4310/jdg/1214429381]

    diff, v16, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeMar 10th 2025
    • (edited Mar 10th 2025)

    added pointer to

    • Tatsuhiko Yagasaki: Homotopy types of Diffeomorphism groups of noncompact 2-manifolds [arXiv:math/0109183, pdf]

    which seems to go a long way towards completing the proof of what is claimed as Thm. 1.14 (p. 43) by Farb & Margalit 12 (who do not provide,as far as I can see, references for the claimed situation with punctures).

    diff, v20, current

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