3
votes
0answers
281 views

$ABA^{-1}B^{-1} = E$ the topology of the space of non-commuting matrices

Is there any discussion of topology of space of matrices $ ABA^{-1}B^{-1} = E $ with $E$ diagonal matrix, $A,B \in \mathrm{GL}(n,\mathbb{C})$? E.g. is this a variety of just a scheme? How many ...
10
votes
3answers
292 views

Topology on the Unitary Dual

Suppose I have a locally compact topological group G. The unitary dual of G is the set of equivalence classes of irreducible unitary representations of G. Now, it seems to me that the sensible way of ...
4
votes
2answers
263 views

Complexification or 'real'ization of Mapping Class group.

So is there a complexification or 'real'ization of the mapping class group or can it be realised as a lattice in some lie group. like $PSL(2, \mathbb Z)$ in $PSL(2, \mathbb R)$. for g=1 this certainly ...
6
votes
2answers
231 views

equivariant cohomology of the complement to the arrangment $\cup_{i\neq j}overrightarrow{x_i} = overrightarrow{x_j}$?

Let $V=\mathbb{R}^d$ be a $d$-dimensional (Euclidian) vector space over real numbers. Let $G=SO(V)$ be a compact Lie group of linear orthogonal transformations of $V$. Let $Conf_n(V)$ be the space of ...
8
votes
0answers
276 views

What is (explicitly) known about the SL(n,C) character varieties of 3-manifolds ?

The $SL(2,{\bf C})$ character variety of a 3-manifold with 1-cusp $M$ (like a knot complement in the 3-sphere) essentially coincide with the variety defined by the A-polynomial. Those polynomials are ...
5
votes
1answer
137 views

Algorithmic Borel Finiteness?

It is a theorem of Borel that there is a finite number of arithmetic hyperbolic manifolds of volume bounded above by $V.$ Is there any algorithm (or hope of an algorithm) to actually construct all of ...
3
votes
1answer
236 views

Kernel of the representation of the mapping class group to $Aut(F_n)$

Let $S_{g,1}$ be a orientable compact surface of genus $g$ with one boundary component and $\Gamma_{g,1}$ the mapping class group. By $F_n$ I denote the free group on $n$ generators. One obtains a ...
8
votes
2answers
368 views

Cohomology of representation varieties

Perhaps this question is too general then I am sorry about this. My question is the following. Let $\pi$ be the fundamental group of a compact surface of genus $g$ (with if necessary $n$ punctures) ...
13
votes
6answers
3k views

Topology of SU(3)

$U(1)$ is diffeomorphic to $S^1$ and $SU(2)$ is to $S^3$, but apparently it is not true that $SU(3)$ is diffeomorphic to $S^8$ (more bellow). Since $SU(3)$ appears in the standard model I would like ...
6
votes
1answer
380 views

What is the image of the half/full twist in the Hecke algebra, in the Kazhdan-Lusztig basis? What is the corresponding complex of Soergel bimodules?

Let $B_n$ be the braid group on $n$ strands. It has generators $\tau_i$ for $i = 1,\ldots,n-1$ which exchange the $i$th and $(i+1)$st strands, and which satisfy the relations $\tau_i \tau_j = \tau_j ...
17
votes
2answers
560 views

Lie algebra automorphisms and detecting knot orientation by Vassiliev invariants

Recall that there are knots in $\mathbf{R}^3$ that are not invertible, i.e. not isotopic to themselves with the orientation reversed. However, it is not easy to tell whether or not a given knot is ...
1
vote
2answers
413 views

Interesting representations/cohomology of surface groups?

For purposes of my own, I'm interested in constructing connected spaces, without recourse to geometric realisation or the like, that have non-trivial homotopy groups in dimension 1 and 2 and are not ...