Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: Gauss-Manin connection

Bottom of Page

1 to 8 of 8

  1.  
    • CommentRowNumber1.
    • CommentAuthorzskoda
    • CommentTimeJul 19th 2011
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJul 25th 2021
    • (edited Jun 8th 2022)

    [ this was nonsense ]

    diff, v4, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 8th 2022

    I have replaced the previously somewhat nonsensical Idea-sentence with something more sensible (but still vague). But mainly I have added a bunch of original and of review references.

    diff, v6, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 19th 2022

    added these pointers:

    diff, v8, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2022
    • (edited Jun 21st 2022)

    Following my own scattered remarks in another thread (here) I have now written a section – here – which means to motivate, explain and spell out what I think is a neat abstract homotopy-theoretic construction of the Gauss-Manin connection on bundles of twisted generalized cohomology groups over fibers of a locally trivial topological fiber bundle.

    This is essentially a category-theoretic elaboration on a couple of informal sentences on Gauss-Manin connections given in Etingof, Frenkel & Kirillov 1998, §7.5, all of which is rather different in style from the algebro-geometric discussion in most of the literature. While the special case here is possibly uninteresting for the algebro-geometer (?) it is rather remarkable in itself in that it yields, in particular, non-trivial solutions to the Knizhnik-Zamolodchikov equation (as recalled in that EFK98, §7.5).

    This is somewhat remarkable once you observe that the evident HoTT-formulation of the model-category theory proof spelled out here is essentially a one-line tautology (the homotopy pullbacks become just substition, the fiberwise 0-truncation requires no further comment, and the Beck-Chevalley argument is nothing but the syntactical compatibility of substitution with dependent product).

    diff, v14, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 22nd 2022

    for completeness, I have added a remark (here) recalling why that right base change of the fiberwise mapping space really is the space of sections expected in the definition of twisted cohomology

    diff, v15, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJun 24th 2022
    • (edited Jun 24th 2022)

    we are making a pdf-writeup of this observation, I have added the pointer:

    diff, v21, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeFeb 1st 2024

    In the first paragraph of “For fiber bundles” (here) I have added links to 1-page pdf-s reproducing the exact pages in Voisin and Etingof et al., respectively, where they give their definition of the GM-connection.

    diff, v26, current

1 to 8 of 8

  1.