### Working paper

## Seifert cohomology of trees

We consider a braid β which acts on a punctured plane. Then we construct a local

system on this plane and find a homology cycle D in its symmetric power, such that D · β(D) coincides with the Alexander polynomial of the plait closure of β.

The Alexander polynomial in several variables is defined for links in three-dimensional homology spheres, in particular, in the Poincaré sphere: the intersection of the surface *S*={(*z*1,*z*2,*z*3)∈C3:(*z*1)5+(*z*2)3+(*z*3)2=0} with the 5-dimensional sphere S*ε*5={(*z*1,*z*2,*z*3)∈C3:|*z*1|2+|*z*2|2+|*z*3|2=*ε*2}. An algebraic link in the Poincaré sphere is the intersection of a germ of a complex analytic curve in (*S*, 0) with the sphere S*ε*5 of radius *ε* small enough. Here we discuss to which extent the Alexander polynomial in several variables of an algebraic link in the Poincaré sphere determines the topology of the link. We show that, if the strict transform of a curve in (*S*, 0) does not intersect the component of the exceptional divisor corresponding to the end of the longest tail in the corresponding *E*8-diagram, then its Alexander polynomial determines the combinatorial type of the minimal resolution of the curve and therefore the topology of the corresponding link. The Alexander polynomial of an algebraic link in the Poincaré sphere is determined by the Poincaré series of the filtration defined by the corresponding curve valuations. (They coincide with each other for a reducible curve singularity and differ by the factor (1−*t*) for an irreducible one.) We show that, under conditions similar to those for curves, the Poincaré series of a collection of divisorial valuations determines the combinatorial type of the minimal resolution of the collection.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.