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A stub (now just recording references and links) graph homology with redirect and section open also for graph cohomology. Related person entry Andrey Lazarev and more references at operad. In operadic literature there is a terminology Feynman transform.
Related entry Rozansky-Witten theory with redirects Rozansky-Witten class, Rozansky-Witten invariant and Alastair Hamilton, one of the researchers on graph (co)homology. Graph cohomology Kontsevich used in his approach to obtain Rozansky-Witten invariants. Variants of Kontsevich’s construction is used in papers of Hamilton and Lazarev to obtain certain characteristic classes of infnity algebras, and apply them to moduli spaces, including to obtain some of the “tautological classes” on compactifications of moduli spaces of curves (Kontsevich also had similar applications from the beginning, but tghe other works show some of the systematic theory and further results).
This sentence here at ribbon graph seems to be broken:
and with a cyclic ordering of (“on”) each vertex.
What do we want to say? That there is a cyclic order on all the edges incident on a given vertex, no?
The structure of a graph complex reflects a structure in the Chevalley-Eilenberg complex of a certain Lie algebra; and the graph homology to the relative Lie homology of that Lie algebra as shown by Kontsevich.
Can you say which Lie algebra that is? It seems natural to wonder how that relates to the Lie algebra of graphs considered by Connes-Kreimer and others (the one used in renormalization).
They interpret it as certain “Lie algebra of noncommutative Hamiltonians”. I can write more about it later. It does not look directly similar to the Connes-Kreimer business.
and with a cyclic ordering of (“on”) each vertex.
Well a vertex is by the definition a set which is element of a partition of the set of all half edges into subsets. One talks about ordering of a set. But regarding the sematics vertex it seems more appropriate to say orientation on each vertex. More descriptive is to say the ordering on the set of half edges incident to a vertex, for all vertices.
cyclic order on all the edges incident on a given vertex
Half-edges, Urs. Because one edge can have both half edges incident to the same vertex so if we order edges incident to a vertex there is an ambiguity.
Zoran, I understand the definition, but I find it sounds confusing. I think it would be good if you added clarification to the entry.
Surely, thanks. Just I need time in managing all. I just got 18 pages letter from my other collaborator and had an appointment with a student :) But I will continue with this circle of entries :)
An interesting paper today: arxiv/1009.1654 "We show that the zeroth cohomology of Kontsevich's graph complex is isomorphic to the Grothendieck-Teichmueller Lie algebra grt."
Today in the archive
*Jian Qiu, Maxim Zabzine, _Introduction to graded geometry, Batalin-Vilkovisky formalism and their applications_, arxiv/1105.2680
I wondered if the characteristic classes like in the articles of Alistair Hamilton with Andrey Lazarev, building on Kontsevich’s original work, can be put in parallel framework with the one used in $n$Lab, via generalized Chern-Weil theory. One strange point was for example the fact that for an analogue of the Chern character they do not take a trace. The values of the characteristic classes were in the graph complex. I did not notice where a Chevalley-Eilenberg complex connects to the graph complex, but it seems the paper above exposes more clearly some details about Kontsevich’s original procedure, cf. e.g. Theorem 6.1 and the discussion at the bottom of 59. I mean nothing new here but I find it written in a way which is more suggestive.
Thanks for alerting me, I’ll try to have a look.
added brief remark on the relation between graph complexes and the real cohomology of configuration spaces (here), so far really just a pointer to
I changed the organization of the first few sections: It used to have a section “Graph homology” which however really was concerned with defining the graph complex, and then “Graph cohomology” which was empty. I removed both these headlines and instead replaced them with “Graph complex”, mentioning in the text now that, of course, the chain homology of the graph complex is referred to as “graph homology”.
The entry needs much more and more substantial work, of course.
I removed the very first paragraph of the entry, as follows:
The first lines were about how “we” will or will not split this entry. Even if this weren’t anachronistic by now, this does not seem the right thing to say in the first line of an entry.
There was a vague indication of which version of Kontsevich’s the entry is going to follow. I moved the line about Kontsevich having several versions to around the pointers to his articles in the References-section and suggest that instead of being vague we next try to be precise and actually state the different definitions in the Definition section, as far as they are of interest.
Finally there was a sentence about obtaining graph complexes from Feynman transforms of modular operads. For the moment I moved this to a new Examples-section. If it deserves to be mentioned right at the beginning in an Idea-section, then it needs more explanation.
added pointer to
because Campos-Willwacher 16 say that this is the origin of the definition of the graph complex which they are using and recalling in their section 3.
(Not sure yet if I recognize Campos-Willwacher’s differential on top of p. 8 in Kontsevich’s definition, his Lemma 3.)
spelled out (here) the definition of the Graph complex via Poincare-duality pairing from Campos-Willwacher 16., allegedly following Kontsevich 99b.
Not sure yet how this is supposed to be related to the other definition, via grafting of graphs.
Choose a linear basis $\{e_i\}_{i \in \{1, 2, \cdots, n\}}$ of $V$ such that $\{e_2\}$ is a linear basis for $\overline{V}$.
Is $\overline{V}$ perhaps supposed to have as basis $\{e_i\}_{i \in \{2, \cdots, n\}}$?
And presumably $N$ rather than $n$.
Thanks once more for proof-reading. Yes, it should (have) read:
Choose a linear basis $\{e_i\}_{i \in \{1, 2, \cdots, N\}}$ of $V$ such that $\{e_2, \cdots, e_N\}$ is a linear basis for $\overline{V}$.
Fixed it now.
So what I spelled out so far is maybe the “pre-graph”-complex, or something. Needs to be expanded to the full thing…
added pointer to
This is finally a good, clean account of the graph complex.
I have removed again my attempted re-statement of the Definition in Campos-Willwacher 16 for compact manifolds, for it just seems too vague and roundabout-way.
So now the Definition-section is back to what it used to be, being implicitly for the case in $\mathbb{R}^d$. But I am not sure if that original definition in our entry is really right or good either.
A decent account of the plain definition is (only?) in Lambrechts-Volić 14, section 6. I have added prominent pointer to that. But I think eventually we should scratch the definition currently in our entry and (re-)write it the way they do.
I realize that I sm not sure how to really parse the second bullet point in Def 6.5 in Lambrechts-Volic arxiv:0808.0457.
I am guessing the statement should be that two graphs that differ only by a permutation of the ordering of their internal vertices represent elements of the graph complex that differ by the sign of that permutation. Equivalently, they differ by a minus sign if a pair of consecutive internal labels is transposed.
Is that what is meant?
Now the full definition is there.
Not proof-read yet.
started a section collecting references on the cohomology of graph complexes: here
I tried to upload here the note on graph cocycles found by computer experiments by Bar-Nata&McKay, transformed from their .ps
to .pdf
My phone thinks it’s uploaded successfully, but my notebook thinks it’s not, but also hangs when I try to make it do it. (?!?)
Your phone is correct.
Ah, thanks. It looks like once again they have the most egregious bugs in Firefox.
I have removed most of what was in the “Properties”-section, because it was repetitive and lacking enough detail to even make much sense
Am working on adding something more substantial. Not done yet. But just added pointer to and graphics from section 3.4 in
which removes what was a major puzzlement for me: They prove that that the “STU-relation”, going back to Figure 8 in
is what characterizes the graph cohomology in the degree $2 \#\! Edges - 3 \#\! Vertices_{int} - \#\!Vertices_{ext} = 0$
The Goodwillie-Weiss manifold calculus seems to have something to say here:
The manifold calculus of Goodwillie and Weiss proposes to reduce questions about embedding spaces of manifolds to questions about mapping spaces of the (little-disks modules of) configuration spaces of points on those manifolds. We will discuss real models for these configuration spaces. Furthermore, we will see that a real version of the aforementioned mapping spaces is computable in terms of graph complexes. In particular, this yields a new tool to study diffeomorphism groups and moduli spaces.
Hmm, something is made interestingly harder by the shift from codimension 3 embeddings to codimension 2 (MO).
So the difference is related to the size of the codimension of the embedding and the extra complications in the knot case arising from maps of the 1d circle into a 3d space, as in the MO answer mentioned in #34?
The difference between the graph complex for configuration spaces of points and that for knot spaces is most minor and easy to miss:
In the latter case, but not in the former, the arcs between consecutive external vertices count as contractible edges for the differential. And external vertices have degree -1 instead of 0. Everything else is the same!
It is conceivable that there is a systematic derivation of graph complex models from Weiss calculus. Volic in his articles talks about relating the two. But if there is such an explanation, I haven’t seen anyone provide any hints for it yet.
I have reworked the Idea-section (here) to bring out the difference between the two similar but different types of graph complexes (one for configuration spaces of points, the other for spaces of knots).
Then I started to write out the precise definition of the latter (here) but didn’t get very far before running out of time now.
Spelled out the example (here) of the first cocycle in $KnotGraphs(\mathbb{R}^3)$ .
This computation convinces me that in the graph complex for knot space cohomology it must be understood that we are to identify graphs that differ by a cyclic ordering of the labels of their external vertices.
(This sounds very obvious, but I don’t see this stated clearly in the literature.)
Ah, there is a mistake. Both of these cocycles actually vanish. Will fix but not tonight.
Surely someone has written down these low dimension cocycles, though I guess there something to be gained in working them out oneself.
Oh, or is it that things haven’t been worked out. I see Willwacher say about one form of Kontsevich graph complex
The cohomology in positive degrees is still unknown.
I have fixed the example of the Peace symbol graph, now here
(It was not wrong before, but it was just zero in the graph complex, due to no internal vertices. )
Surely someone has written down these low dimension cocycles,
One would think so, wouldn’t one.
There are two concrete examples spelled out in Figures 2 and 3 in Cattaneo, Cotta-Ramusino, Longoni02. I have reproduced the computation here.
Doing so made me realize that the definition of the graph complex on their p. 18 is a little problematic. Where they say “We assume that the labelling of the external vertices is cyclic” I think what they must mean is that graphs are to be identified whose external labels differ by a cyclic permutation.
In any case, this is for another version of the graph complex, the one that in the entry is now called $KnotGraphs(\mathbb{R}^3)$ (here).
Here, I am instead after examples of cocycle in what in the entry is denoted $Graphs_n(\mathbb{R}^3)$, which is the complex modelling configuration spaces.
$\,$
Many examples are spelled out in Bar-Natan and McKay.
But what are these examples of? The complex they work in looks like the one called $Graphs_0(\mathbb{R}^2)$ in the entry. But this is supposed to be quasi-isomorphic to $\Omega^\bullet( Conf_0(\mathbb{R}^2) ) \simeq \Omega^\bullet( \ast ) \simeq \mathbb{R}$, so that there shouldn’t be any non-trivial examples in this case.
Indeed, as far as I see, Bar-Natan and McKay consider further filtering/bidegrees on this complex, and their examples are cocycles in particular filtering stage only. But then, what does this all mean. (?)
$\,$
I see Willwacher say …
I am still working on figuring out which exact version of the graph complex is meant in these articles. I think in these statements what is mostly meant is $KnotGraphs(\mathbb{R}^3)$, though I feel uncertain about this. In other articles it must be $Graphs(\Sigma)$ instead, but the precise details of the definition considered are not easy to extract (for me, at least).
$\,$
A bit of a mess, to my mind. Maybe some expert sees me struggling here and lends a hand. I’d be grateful!
A bit of a mess, to my mind.
Yes! This paper has a load of variants $GC$, $fGC$, $dfGC$, each of which may have annotations to show that it is ’uncompleted’, ’1-vertex irreducible’, ’connected’.
You’d think someone in the field would look to standardize language here. Maybe a MathOverflow question is called for.
Not to mention hairy graphs.
So, da full graph complex $dfGC$ and its siblings, relating to Grothendieck-Teichmueller, have anti-commuting edges and commuting vertices, like the graph complexes $Graphs_n(\mathbb{R}^{2D})$ modelling configuration spaces in even dimensions do. However, they don’t have any distinction between external and internal vertices. Hence if (their linear dualization) were anything like these graph models for configuration spaces, it would have to be $Graphs_0(\mathbb{R}^{2D})$ (zero external vertices) which however is trival. Also, it doesn’t look like the dualization of the differential in da full graph complex is the one appropriate for modeling configuration spaces and knot spaces.
So this must be an entirely different, third, class of graph complexes altogether.
Those ’hairy’ ones (#47) have external vertices and are used for embeddings and knot invariants.
Sure , graph complexes with external edges do exist, modelling embeddings. That’s what I currently have in the entry, at least for some cases. The question is if they related to the da graph complex computing the Grothdieck-Teichmuller Lie algebra, or if the latter is a third kind of graph complex altogether.
Not sure why you’re writing ’da’.
’d’ in ’dfGC’ is ’directed’.
They write
“the full directed graph complex dfGC”
With your interpretation, it should be “fdGC” instead. So this can’t be how they want this to be pronounced. ;-)
added this useful pointer:
Theorem 9.3 there directly identifies the rational homotopy groups of the configuration space in terms of the cohomology of indecomposable graphs
Re #44 and elsewhere: not actually the Peace Symbol. More like the Mercedes-Benz logo.
True. I found one occurrence of Peace in the entry, have changed it to “minimal trivalent diagram”.
What bothers me more is that this diagram, being closed (here), must be exact, since there is not supposed to be any cohomology in its degree.
Do you see a trivializing coboundary?
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