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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 4th 2010

    started monodromy

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeFeb 4th 2010

    Now I am confused about the higher homotopy groups aspect. Isn't it that in a sense higher analogues of universal covering spaces are played by higher Postnikov fibers ? Now monodromy is usually looked for pi-1 case, and it is clear to me why you go to infinity analogue right away. But I do not see how now levels for finite pi-k, or H-k (Hurewicz! once you are there) seen at finite levels are packed into the infinity picture ? Is it worthy to look that way or I have just baroque reminiscences of unwanted analogies ?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeFeb 4th 2010

    Ahm, not sure. Maybe I don't quite understand what you have in mind.

    Did you look at Toen's article? He talks about this oo-monodromy group.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeFeb 4th 2010

    Never mind.

  1. Does a fibration π:EX\pi:E\to X of (nice) topological spaces induce a fibration Π(E)Π(X)\Pi(E)\to \Pi(X)? (this shoud be obviously true or obviously false, but as worn out as I am at the moment I will leave it as a question). If that is true, then according to Toen’s equivalence, one should have that π:EX\pi:E\to X defines a local system on XX. Moreover, since any map f:YXf:Y\to X can be replaced by an equivalent fibration, one should have a good notion of (higher) monodromy defined for any morphism of topological spaces f:YXf:Y\to X.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 28th 2012
    • (edited Jun 28th 2012)

    Does a fibration π:E→X of (nice) topological spaces induce a fibration Π(E)→Π(X)?

    Yes, Π=Sing\Pi = Sing is a right Quillen functor. See here.

    one should have that π:E→X defines a local system on X.

    Yes.

    Moreover, since any map f:Y→X can be replaced by an equivalent fibration, one should have a good notion of (higher) monodromy defined for any morphism of topological spaces f:Y→X.

    Yey, every morphism EXE \to X with κ\kappa-small homotopy fibers is classified by a “monodromy map”

    X FBAut(F) X \to \coprod_{F} B Aut(F)

    where the coproduct ranges over κ\kappa-small homotopy types. If XX is connected we can restrict to one of these and have that EXE \to X is classified by

    XBAut(F). X \to B Aut(F) \,.

    This is originally a theorem of Stasheff and May. But it is also a simple instance of the general statement in section 4 of NSSa.

    • CommentRowNumber7.
    • CommentAuthordomenico_fiorenza
    • CommentTimeJun 28th 2012
    • (edited Jun 28th 2012)

    Thanks! I was interested in this since a linear representation of the monodromy map above is at the heart of the quantization map in section 8 of Topological QFTs from compact Lie groups

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJul 2nd 2017

    I have added to monodromy an elementary point-set discussion of the monodromy of covering spaces, here.

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