Abstract
A complex hyperplane arrangement A is said to be decomposable if there are no elements in the degree 3 part of its holonomy Lie algebra besides those coming from the rank 2 flats. When this purely combinatorial condition is satisfied, it is known that the associated graded Lie algebra of the arrangement group G decomposes (in degrees greater than 1) as a direct product of free Lie algebras, and all the nilpotent quotients of G are combinatorially determined. We show here that the Alexander invariant of G also decomposes as a direct sum of “local” invariants. Consequently, the degree 1 cohomology jump loci of the complement of A have only local components, and the algebraic monodromy of the Milnor fibration is trivial in degree 1.
Topological Invariants of Groups
For a group it is possible to define a lower central series. It is interesting because every group is normal in the previous. The quotient is always abelian and have ranks.
We can define a lie algebra, where the addition is the group composition and the multiplication is given by the commutator. The lower central series gives a natural grading of the studied object.
The Chen Lie Algebra is another kind of Algebra defined from a group. Examples. For the free group the Lie Algebras are particularly easy to calculate.
A different kind of Lie Algebra is defined out of homological data of a group. This gives a connection between the cup product and the lie Algebra.
Ressource: “Browns Book on Homology of Groups”.
Holonomy Lie Algebra
We continue defining Lie Algebra. We come across Malcev Lie Algebra as well as a new object defined from the usual Lie algebra.
Alexander Invariant
Let be the first Element of the lower Central series. e can make a extension relating , , . The alexander invariant This invariant generalizes the usual Alexander polynomial. It has the structure of a special module.
The Alexander Invariant is functorial, mapping from Groups to the mentioned special modules.
Using Lie algebras we can define a infinitesimal Alexander Invariant. There is an easy way to relate both invariants by taking a completion. We can show that this results in isomorphic structures.
Characteristic Varieties
For a f.g. group we define the so called Character Group. The looked at concept is quite concept but interesting since it has uses for Free Groups, Fundamental Groups of Reimann surfaces and Fundamental Groups of Knot komplement.
Resonance Invariant
We look at the cohomology algebra of a finitely generated group. Elements of the first Homology Group define a chain complex by multiplication. Using this we define a resonance variety.
Hyperplane Arrangements
A hyperplane arrangement is a set of (affine) hyperplanes in a complex vector space. We can create an intersection lattices, which is a partially ordered set.
Examples:
- There is a connection to braids!!!! This can be done by looking at all diagonal hyperplanes in .