## DIMACS TR: 94-13

## Combinatorial Bases in Systems of Simplices and Chambers

On the Fundamental Group of the Complement of a Complex
Hyperplane Arrangement

### Authors: Tatiana V. Alekseyevskaya, Grigori Rybnikov

**
ABSTRACT
**

Combinatorial Bases in Systems of Simplices and Chambers
We introduce the notion of a combinatorial basis. Combinatorial bases of
chambers can be described in terms of a game. We describe the algorithms
of decomposition of a convex polytope into shells. In the case of the affine
plane, using the game and the algorithm we construct combinatorial basis
B of chambers. Using the algorithm, we also construct a basis B' of
simplices that together with the basis B of chambers form a "triangular
pair".

On the Fundamental Group of the Complement of a Complex Hyperplane Arrangement

We construct two combinatorially equivalent line arrangements in the complex
projective plane such that the fundamental groups of their complements are
not isomorphic. In the proof we use a new invariant of the fundamental group
of the complement of a line arrangement with prescribed combinatorial type
with respect to isomorphisms inducing the canonical isomorphism of first
homology groups.

Paper available at:
ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1994/94-13.ps

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