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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.
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