The Topology of the Configuration of Projective Subspaces in a Projective Space I

We assume that there is given a finite family of protective subspaces in certain projective space. Our aim is to prove that the simple homotopy type of the union of all the subspaces in question is completely determined by the complex of the nerves resulting from the family, equipped with the filtration obtained by assigning to each simplex the Krull dimension of the corresponding intersection. This fact enables us to compute the homology groups of the union and its complement in principle