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New stub, Gauss-Manin connection.
added these pointers:
Daniel C. Cohen, Peter Orlik, Gauss-Manin Connections for Arrangements, I Eigenvalues, Compositio Math. 136 (2003) 299-316 $[$arXiiv:math/0105063, doi:10.1023/A:1023262022279$]$
Daniel C. Cohen, Peter Orlik, Gauss-Manin connections for arrangements, II Nonresonant weights, Amer. J. Math. 127 (2005) 569-594 $[$arXiv:math/0207114, jstor:40067930$]$
Daniel C. Cohen, Peter Orlik, Gauss-Manin connections for arrangements, III Formal connections, Trans. Amer. Math. Soc. 357 (2005) 3031-3050 $[$arXiv:math/0307210, doi:10.1090/S0002-9947-04-03621-9$]$
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).
we are making a pdf-writeup of this observation, I have added the pointer:
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