  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.
