Let be a vector space over a field . A hyperplane arrangment in is a family of affine hyperplanes in . If all of the hyperplanes pass through 0, is called central; otherwise, it is affine. More generally, a subspace arrangement is a family of affine subspaces of . The same distinction between central and affine subspace arrangement holds.

Example 1   Let . Then the family

of -dimensional subspaces of is a central subspace arrangement, the projective space of dimension over .

Instead of considering all lines through a vector space, we could consider all -dimensional subspaces of the space.

Example 2   Again let , and suppose . Then the family

of -dimensional subspaces of is a central subspace arrangement, the Grassmannian. Observe that .

If is a topological vector space and is a hyperplane arrangement, then it makes sense to ask for the fundamental group of the complement .

Example 3   If is a finite hyperplane arrangement over , then the arrangement partitions into a finite number of contractible cells. By selecting a point in each cell and taking the convex hull of the result, we obtain a polytope combinatorially equivalent to the zonotope dual to the arrangement. Since the question of the fundamental group here is not interesting, we could also use the embedding to complexify . In this case the complement usually has nontrivial fundamental group.

Bibliography

1

Klain, D. A., and G.-C. Rota, , Introduction to geometric probability, Cambridge University Press, 1997.

2

Orlik, P., and H. Terao, Arrangements of hyperplanes, Springer-Verlag, 1992.