Falk's φ₃ Invariant For A Class Of Graphic Arrangements And Line Arrangements

In 1989, M. Falk defined an invariantφ3, for hyperplane arrangements,which is the rank of the abelian group (G3/G4), where G3, G4 are the lower central sequence of fundamental group G=G1 of the complement of the hyperplane arrangement. In 2001, M. Falk proposed the following open problem.Give a combinatorial interpretation ofφ3.And he points out that the problem remains open, even for graphic matroids.In this thesis, we study Falk'sφ3 invariant for graphic arrangements and line arrangements in projective plane and obtained the following result. If a simple graph does not contain the complete graph K4 as a subgraph,φ3 equals to two times of the number of circuits with 3 edges. That answers the question of Falk partially. And a similar statement is true for line arrangements in projective plane.By optimizing the algorithm in the references, we formulate an algorithm for computingφ3 invariant, and proved the formulaφ3=2#C3 for simple graphic arrangements.By a therom due to Zariski, the computing of the fundamental group of the complement of varieties in higher dimensional space can be reduced to computing that of the complement of plane cuves. For this reason we studyφ3 invariant for line arrangements in projective plane. We obtain the following result. The invariantφ3 of line arrangements in projective plane is equal to twice of #C3.