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1. • CommentRowNumber1.
   • CommentAuthordomenico_fiorenza
   • CommentTimeDec 23rd 2011
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   Author: domenico_fiorenza
   Format: MarkdownItexGiven a Lie group $G$ with Lie algebra $\mathfrak{g}$, one can think of a $\mathfrak{g}$-valued 1-form $\omega$ on a differential manifold $M$ as a connection form on the trivial principal $G$-bundle $G\times M\to M$. The curvature of $\omega$ is a $\mathfrak{g}$-valued 2-form on $M$, and $\omega$ is flat precisely when its curvature vanishes. These are in a sense the extreme cases: on the one side the curvature 2-form takes values in the whole of $\mathfrak{g}$, on the other side the curvature takes values in the $0$ Lie subalgebra of $\mathfrak{g}$. I was therefore wondering whether given a Lie subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ there is a meaningful notion of connection form $\omega$ with $\mathfrak{h}$-constrained curvature, i.e. whose curvature 2-form is $\mathfrak{h}$-valued. This should be (hopefully) related to the homogeneous space $G/H$, where $H$ is the Lie subgroup of $G$ corresponding to the Lie algebra $\mathfrak{h}$.

   Given a Lie group with Lie algebra , one can think of a -valued 1-form on a differential manifold as a connection form on the trivial principal -bundle . The curvature of is a -valued 2-form on , and is flat precisely when its curvature vanishes. These are in a sense the extreme cases: on the one side the curvature 2-form takes values in the whole of , on the other side the curvature takes values in the Lie subalgebra of . I was therefore wondering whether given a Lie subalgebra of there is a meaningful notion of connection form with -constrained curvature, i.e. whose curvature 2-form is -valued. This should be (hopefully) related to the homogeneous space , where is the Lie subgroup of corresponding to the Lie algebra .

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 23rd 2011
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   Author: Urs
   Format: MarkdownItexWhat you say reminds me a bit of [[Cartan connections]].

   What you say reminds me a bit of Cartan connections.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeDec 23rd 2011
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   Author: David_Corfield
   Format: MarkdownItexHave you read John Baez on [torsoroids](http://golem.ph.utexas.edu/category/2007/07/derek_wise_on_cartan_geometry.html#c010785)?

   Have you read John Baez on torsoroids?

4. • CommentRowNumber4.
   • CommentAuthordomenico_fiorenza
   • CommentTimeDec 23rd 2011
   • (edited Dec 23rd 2011)
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   Author: domenico_fiorenza
   Format: MarkdownItex@David: I wasn't aware of that. Thanks, that really seems something I have to investigate better. @Urs: I'll have to look into that, too, thanks. What I'm after, as you can have imagined, is a Lie algebra valued forms local description of $G/H$-bundles induced by a G-princiapal bundle which parallels our local description of principal $G$-bundles in terms of $\mathfrak{g}$-valued differential forms on $\Delta^n\times U$ which are flat in the simplicial direction. I that can be done, then it would be another step in reproducing Brylinski-McLaughlin construction of differential characteristic classes via maps from $\Delta^n\times U$ into homogeneous spaces in our Lie algebra integration language.

   @David: I wasn’t aware of that. Thanks, that really seems something I have to investigate better.

   @Urs: I’ll have to look into that, too, thanks. What I’m after, as you can have imagined, is a Lie algebra valued forms local description of -bundles induced by a G-princiapal bundle which parallels our local description of principal -bundles in terms of -valued differential forms on which are flat in the simplicial direction. I that can be done, then it would be another step in reproducing Brylinski-McLaughlin construction of differential characteristic classes via maps from into homogeneous spaces in our Lie algebra integration language.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeDec 23rd 2011
   • (edited Dec 23rd 2011)
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   Author: Urs
   Format: MarkdownItexIt is of course possible to consider the sub-simplicial presheaf of $\exp(\mathfrak{g})_{conn}$ for which the curvature forms satisfy an extra constraint. But now I am not sure what you have in mind with this. To me the problem we'd have to solve is how to get a resolution of $\mathbf{B}G$ by a simplicial object of homology-chains in $G/H$.

   It is of course possible to consider the sub-simplicial presheaf of for which the curvature forms satisfy an extra constraint. But now I am not sure what you have in mind with this. To me the problem we’d have to solve is how to get a resolution of by a simplicial object of homology-chains in .

6. • CommentRowNumber6.
   • CommentAuthorjim_stasheff
   • CommentTimeDec 23rd 2011
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   Author: jim_stasheff
   Format: Text@Urs: how to get a resolution of BG by a simplicial object of homology-chains in G/H. What kind of resolution? Do you have in mind H --> G --> G/H --> BH --> BG ??

   @Urs: how to get a resolution of BG by a simplicial object of homology-chains in G/H.
   
   What kind of resolution?
   Do you have in mind H --> G --> G/H --> BH --> BG ??
7. • CommentRowNumber7.
   • CommentAuthordomenico_fiorenza
   • CommentTimeDec 23rd 2011
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   Author: domenico_fiorenza
   Format: MarkdownItex@Urs: what I have in mind is not a subsheaf of $exp(\mathfrak{g})_{conn}$ but a supersheaf: in $exp(\mathfrak{g})_{conn}$ we impose curvature to vanish in the vertical direction, and this is the origin of the topological constrain. Namely, to give a flat $\mathfrak{g}$-connection on $\Delta^n$ is the same thing as giving a based map from $\Delta^n$ to $G$. This means that if we are given a flat $\mathfrak{g}$-connection on $\partial \Delta^n$, the obstruction to extending it to a flat $\mathfrak{g}$-connection on $\Delta^n$ is a topological one: it is an element in $\pi_n(G,e)$. Yet, assume we weaken our request and ask to extend the flat $\mathfrak{g}$-connection on $\partial \Delta^n$ to an $\mathfrak{h}$-curvature constrained connection on $\Delta^n$. Maybe this can be done and if we are lucky the obstruction to this is an element in $\pi_n(G/H,[e])$. Surely thsi would not solve the issue with Pontryagin classes (since there is still a finite group there whicih is not seen by these Lie theoretic construtions), but this could hopefully handle Chern classes for princiapal $U(n)$-bundles (the Stiefel manifolds $U(n)/U(k)$ are connected-to-the-point-one-is-interested-in). @jim: welcome in this discussion: I really hoped you would have joined here

   @Urs: what I have in mind is not a subsheaf of but a supersheaf: in we impose curvature to vanish in the vertical direction, and this is the origin of the topological constrain. Namely, to give a flat -connection on is the same thing as giving a based map from to . This means that if we are given a flat -connection on , the obstruction to extending it to a flat -connection on is a topological one: it is an element in . Yet, assume we weaken our request and ask to extend the flat -connection on to an -curvature constrained connection on . Maybe this can be done and if we are lucky the obstruction to this is an element in . Surely thsi would not solve the issue with Pontryagin classes (since there is still a finite group there whicih is not seen by these Lie theoretic construtions), but this could hopefully handle Chern classes for princiapal -bundles (the Stiefel manifolds are connected-to-the-point-one-is-interested-in).

   @jim: welcome in this discussion: I really hoped you would have joined here

8. • CommentRowNumber8.
   • CommentAuthorjim_stasheff
   • CommentTimeDec 24th 2011
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   Author: jim_stasheff
   Format: Text@Domenico:don't know if it's relevant, but just before Serre's thesis diverted attention from de Rham cohomology, Cartan et Cie were working on bundles with G/H fibers

   @Domenico:don't know if it's relevant, but just before Serre's thesis diverted attention from de Rham cohomology,
   Cartan et Cie were working on bundles with G/H fibers
9. • CommentRowNumber9.
   • CommentAuthordomenico_fiorenza
   • CommentTimeDec 27th 2011
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   Author: domenico_fiorenza
   Format: MarkdownItexI got an excellent answer from Robert Bryant <a href="http://mathoverflow.net/questions/84218/lie-algebra-valued-1-forms-and-pointed-maps-to-homogeneous-spaces">here</a>

   I got an excellent answer from Robert Bryant here

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeDec 28th 2011
    • (edited Dec 28th 2011)
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    Author: Urs
    Format: MarkdownItexHi Domenico, sorry for being silent for so long. I was utterly exhausted after the last two months and needed a good break in the last week, just family, and offline :-) Hope you all had good Christmas holidays. But now some thoughts. First: it seems to me that for the construction that we need, the question of how to most suitably parameterize maps into a coset is secondary. The primary problem is to exhibit a useful resolution of $\mathbf{B}G$ in terms of _homology_ cycles in cosets, whichever way these are described. Since we are discussing this out in the open anyway, let me briefly recall for other readers what this is all about, since otherwise this discussion must seem mysterious to everybody else: **Context**. In * Brylinski, McLaughlin, _Čech cocycles for characteristic classes_ ([EUCLID](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1104286562)) the authors constructed what the title announces: [[Cech cohomology]] representatives of all classical [[characteristic classes]]. In * Fiorenza, Schreiber, Stasheff, _Čech cocycles for differential characteristic classes_ ([[schreiber:Cech Cocycles for Differential characteristic Classes|nLab]]) we observed that _for degree-4 classes_ this construction naturally refines to a morphism of smooth higher moduli stacks $\mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn}$ from the moduli stack of $G$-principal connections to that of [[circle n-bundle with connection|circle 3-connections]]. More generally, for any degree $(n+1)$-class, this refinement generalizes to a "[[infinity-Chern-Weil theory|higher Chern-Weil homomorphism]]" of this form where, however, $G$ is now required to be an _[[infinity-group|n-group]]_ that extends the original Lie group. For instance this way the second [[Pontryagin class]] is obtained not as a morphism on the moduli stack of $\mathrm{Spin}$-connections, but on that of [[string 2-group|String]]-connections $\frac{1}{6} \hat \mathbf{p}_2 : \mathbf{B} String_{conn} \to \mathbf{B}^7 U(1)_{conn}$. Therefore there is the evident **Open question** Can the construction of Brylisnki-MacLaughlin be refined to a morphism of smooth higher moduli stacks $\mathbf{B}G_{conn} \to \mathbf{B}^{n} U(1)_{conn}$ _with $G$ a Lie group_ also for the case of higher characteristic classes? Or in other words, can we "push down" the universal construction of Fiorenza-Schreiber-Stasheff to lower connected covers? For instance, can we find the extension $\hat \mathbf{p}_2$ in $$ \array{ \mathbf{B} String_{conn} &\stackrel{\frac{1}{6}\hat \mathbf{p}_2}{\to}& \mathbf{B}^7 U(1)_{conn} \\ \downarrow && \downarrow_{\cdot 6} \\ \mathbf{B} Spin_{conn} &\stackrel{\hat \mathbf{p}_2}{\to}& \mathbf{B}^7 U(1)_{conn} } \,. $$ ?? (We have the corresponding extension on the level of topological universal characteristic maps. The task is here to first refine to smooth higher stacks, and then further to their differential refinement.) I want to state an attempt to answer this question. But I'll do so in the next comment.

    Hi Domenico,

    sorry for being silent for so long. I was utterly exhausted after the last two months and needed a good break in the last week, just family, and offline :-) Hope you all had good Christmas holidays.

    But now some thoughts.

    First: it seems to me that for the construction that we need, the question of how to most suitably parameterize maps into a coset is secondary. The primary problem is to exhibit a useful resolution of in terms of homology cycles in cosets, whichever way these are described.

    Since we are discussing this out in the open anyway, let me briefly recall for other readers what this is all about, since otherwise this discussion must seem mysterious to everybody else:

    Context.

    In

    • Brylinski, McLaughlin, Čech cocycles for characteristic classes (EUCLID)

    the authors constructed what the title announces: Cech cohomology representatives of all classical characteristic classes.

    In

    • Fiorenza, Schreiber, Stasheff, Čech cocycles for differential characteristic classes (nLab)

    we observed that for degree-4 classes this construction naturally refines to a morphism of smooth higher moduli stacks from the moduli stack of -principal connections to that of circle 3-connections.

    More generally, for any degree -class, this refinement generalizes to a “higher Chern-Weil homomorphism” of this form where, however, is now required to be an n-group that extends the original Lie group.

    For instance this way the second Pontryagin class is obtained not as a morphism on the moduli stack of -connections, but on that of String-connections .

    Therefore there is the evident

    Open question

    Can the construction of Brylisnki-MacLaughlin be refined to a morphism of smooth higher moduli stacks with a Lie group also for the case of higher characteristic classes?

    Or in other words, can we “push down” the universal construction of Fiorenza-Schreiber-Stasheff to lower connected covers? For instance, can we find the extension in

    ??

    (We have the corresponding extension on the level of topological universal characteristic maps. The task is here to first refine to smooth higher stacks, and then further to their differential refinement.)

    I want to state an attempt to answer this question. But I’ll do so in the next comment.

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeDec 28th 2011
    • (edited Dec 28th 2011)
    • PermaLink
    Author: Urs
    Format: MarkdownItexHi Domenico, let me try to play around with a suggestion for how to proceed. Let $G = Spin(n)$. We need a simplicial presheaf resolution $Q \stackrel{\simeq}{\to} G^{\times \bullet} =: \mathbf{B}G_{ch}$ that supports that BMcL-construction not just for $k = 1$ but also for higher $k$. For $k = 1$ we used $$ Q := \mathbf{cosk}_3 \exp(\mathfrak{so}(n)) $$ which we may think of as "$\mathbf{cosk}_3 Sing_{based, smooth}(G)$", the simplicial complex of smooth simplices in $G$ that are based at the neutral element. The [[simplicial skeleton|coskeleton]] construction at some point does away with the singular simplices and fills up everything with unique abstract simplices. Our task is: replace the blunt coskeleton construction by a construction that instead fills in all higher simplicial spheres by homology chains in the coset $Spin(n)/Spin(q)$. There is an obvious guess for how to do this: Let $Q$ be given as follows * a single object; * 1-morphisms are smooth based paths in $Spin(n)$; * 2-morphisms are smooth based triangles in $Spin(n)$; now: * 3-morphisms are smooth based 3-chains over $\mathbb{R}$ in $Spin(n)/Spin(q)$, whose faces have representatives by maps $\Delta^2 \to Spin(n)$; * 4-morphisms are smooth based 4-chains over $\mathbb{R}$ whose faces are of the above form and so on until * $(4k-2)$-morphisms are smooth based $(4k-2)$-chains etc. and then complete by the coskeleton * there are unique higher cells filling any bounding ball. The evident projection $Q \to \mathbf{B}G_{ch}$ should be a weak equivalence, since, by that table on p. 3 of BMcL, the integral homology of the coset space is all torsion in the relevant range, and hence vanishes over the reals. Finally, $Q$ should have a differential refinement $Q_{conn}$ and the BMcL-construction should extend to a morphism of simplicial presheaves $\hat \mathbf{p}_k : Q_{conn} \to \mathbf{B}^{4k-1} U(1)_{conn}$. I'd hope. That would be my suggestion. I haven't fully thought this through. I am saying this in order to indicate in which direction I think we should be pushing. [edit: after a bit of fiddling, I am not sure if I can make this work...]

    Hi Domenico,

    let me try to play around with a suggestion for how to proceed.

    Let . We need a simplicial presheaf resolution that supports that BMcL-construction not just for but also for higher .

    For we used

    which we may think of as “”, the simplicial complex of smooth simplices in that are based at the neutral element. The coskeleton construction at some point does away with the singular simplices and fills up everything with unique abstract simplices.

    Our task is: replace the blunt coskeleton construction by a construction that instead fills in all higher simplicial spheres by homology chains in the coset .

    There is an obvious guess for how to do this:

    Let be given as follows

    • a single object;

    • 1-morphisms are smooth based paths in ;

    • 2-morphisms are smooth based triangles in ;

    now:

    • 3-morphisms are smooth based 3-chains over in , whose faces have representatives by maps ;

    • 4-morphisms are smooth based 4-chains over whose faces are of the above form

    and so on until

    • -morphisms are smooth based -chains etc.

    and then complete by the coskeleton

    • there are unique higher cells filling any bounding ball.

    The evident projection should be a weak equivalence, since, by that table on p. 3 of BMcL, the integral homology of the coset space is all torsion in the relevant range, and hence vanishes over the reals.

    Finally, should have a differential refinement and the BMcL-construction should extend to a morphism of simplicial presheaves . I’d hope.

    That would be my suggestion. I haven’t fully thought this through. I am saying this in order to indicate in which direction I think we should be pushing.

    [edit: after a bit of fiddling, I am not sure if I can make this work…]

12. • CommentRowNumber12.
    • CommentAuthordomenico_fiorenza
    • CommentTimeDec 29th 2011
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    Author: domenico_fiorenza
    Format: MarkdownItexHi Urs, thanks for this additions and suggestions. I'm now away for winter holidays and so I will be for other 4-5 days. After that I'll come back to this. But as you see I'm giving a glance to what's happening on nforum, so if this discussions evolves while I'm out I'll nevertheless be able to follow it at least a bit

    Hi Urs,

    thanks for this additions and suggestions. I’m now away for winter holidays and so I will be for other 4-5 days. After that I’ll come back to this. But as you see I’m giving a glance to what’s happening on nforum, so if this discussions evolves while I’m out I’ll nevertheless be able to follow it at least a bit