best,

Danny

(ps I haven’t updated my blog in a long time; I keep intending to get around to it though)

I’ve been reading about the Goldman form a little bit recently and I found this blog post, which was very nice to read. I had what I assume is a stupid question: In a couple of his papers Goldman restricts to working with closed surfaces of genus g > 1. Do you know why he doesn’t include the torus? My impression is that it has something to do with being abelian, but I don’t know what. Do you know if he has this restriction because the symplectic form (or corresponding Poisson bracket) don’t exist, or because the character variety isn’t as well-behaved?

A paper that I thought was interesting that uses this Poisson bracket is http://www.ams.org/mathscinet/search/publdoc.html?pg1=MR&s1=1691437&loc=fromreflist
A one-sentence summary is that the “deformation quantization of the ring of functions on the character variety of of a surface” has a natural geometric interpretation.

Peter
p.s. I was a student in your class on complex curves a few years back (winter of 04-05?), which was a nice class :-)

There’s an interesting earlier (ie 1978) paper of
Turaev entitled “Intersections of loops in
two-dimensional manifolds”. In that paper, he gives
an intersection pairing on of a surface (with
one boundary component; the basepoint is on the boundary) that
is superficially very similar to the Goldman bracket,
but actually has rather different, intriguing properties.

It is a pairing
.
Unlike the Goldman bracket, its values depend strongly on
the basepoint. However, it interacts very well with
the group-theory of the surface group. In particular,
it is actually a biderivation in an appropriate sense. Even
more interestingly, I noticed when I read Turaev’s paper that
it interacts well with the derived series of . Let
be the quotient of by the kth term
of its derived series, so is a k-step solvable
group. Then Turaev’s bracket descends to a pairing
. For ,
we have and this is just the algebraic intersection
pairing. For the higher terms, however, we get something new. In
particular, for we have equal to the
first homology group of the surface, so we get an “intersection pairing”
on the two-step solvable truncation of with values in
the group ring of its first homology group!

I’ve always thought that this should have nice applications
(maybe to understanding the “solvable” version of the Johnson
filtration of the mapping class group), but I haven’t yet managed
to find any…

It is a pairing
.
Unlike the Goldman bracket, its values depend strongly on
the basepoint. However, it interacts very well with
the group-theory of the surface group. In particular,
it is actually a biderivation in an appropriate sense. Even
more interestingly, I noticed when I read Turaev’s paper that
it interacts well with the derived series of . Let
be the quotient of by the kth term
of its derived series, so $\latex S_k$ is a k-step solvable
group. Then Turaev’s bracket descends to a pairing
. For ,
we have and this is just the algebraic intersection
pairing. For the higher terms, however, we get something new. In
particular, for we have equal to the
first homology group of the surface, so we get an “intersection pairing”
on the two-step solvable truncation of with values in
the group ring of its first homology group!

I’ve always thought that this should have nice applications
(maybe to understanding the “solvable” version of the Johnson
filtration of the mapping class group), but I haven’t yet managed
to find any…