A long exact sequence in cohomology for deleted and restricted subspaces arrangements

TL;DR

This paper extends the classical long exact sequence relating cohomology of hyperplane arrangement complements to arbitrary subspace arrangements, providing new tools for understanding their topological invariants.

Contribution

It introduces a long exact sequence in rational cohomology for arbitrary subspace arrangements, generalizing known results from hyperplane arrangements.

Findings

Established a long exact sequence connecting cohomologies of arrangements and their deletions and restrictions.

Derived new relations between Betti numbers and Poincare polynomials for subspace arrangements.

Extended classical formulas from hyperplane arrangements to more general subspace arrangements.

Abstract

The notions of deleted and restricted arrangements have been useful in the study of arrangements of hyperplanes. If A is an arrangement of hyperplanes, x in A and A', A'' the deleted and restricted arrangements, there is a formula connecting the Poincare polynomials of the complement spaces M(A), M(A') and M(A''). In this paper, we consider the extension of this formula to arbitrary subspaces arrangements. The main result is the existence of a long exact sequence connecting the rational cohomology of M(A), M(A') and M(A''). Using this sequence, we obtain new results connecting the Betti numbers and Poincare polynomials of deleted and restricted arrangements.