hyperplane arrangement

Let $V$ be a vector space over a field $𝕂$. Ahyperplane arrangment in $V$ is a family$𝒜={\{{\mathscr{H}}_i\}}_{i\in I}$of affine hyperplanes in $V$. If all of the hyperplanespass through $0$, $𝒜$ is called central;otherwise, it is affine. More generally, asubspace arrangement is a family of affine subspaces of $V$.The same distinction between central and affine subspace arrangementholds.

Example 1.

Let $V\mathrm{=}{\mathrm{K}}^n$. Then the family

$$𝕂P^n=\{S\subset V\mid {dim}_{𝕂}(S)=1\}$$

of $\mathrm{1}$-dimensional subspaces of $V$ is a central subspacearrangement, the projective space of dimension $n$ over$\mathrm{K}$.

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

Example 2.

Again let $V\mathrm{=}{\mathrm{K}}^n$, and suppose $\mathrm{0}\mathrm{\le }k\mathrm{\le }n$. Then thefamily

$$\mathrm{Gr}(V,k)=\{S\subset V\mid {dim}_{𝕂}(S)=k\}$$

of $k$-dimensional subspaces of $V$ is a central subspacearrangement, the Grassmannian. Observe that$\mathrm{K}\mathit{}{P}^n\mathrm{=}\mathrm{Gr}\mathit{}\mathrm{(}{\mathrm{K}}^n\mathrm{,}\mathrm{1}\mathrm{)}$.

If $V$ is a topological vector space and $𝒜$ is ahyperplane arrangement, then it makes sense to ask for thefundamental group of the complement$V\setminus {\bigcup }_{\mathscr{H}\in 𝒜}\mathscr{H}$.

Example 3.

If $\mathrm{A}$ is a finite hyperplane arrangement over$V\mathrm{=}{\mathrm{R}}^n$, then the arrangementpartitions (http://planetmath.org/Partition) $V$into a finite number of contractible cells. By selectinga point in each cell and taking the convex hull of the result,we obtain a polytope combinatorially equivalent to thezonotope dual to the arrangement. Since the question ofthe fundamental group here is not interesting, we could alsouse the embedding ${\mathrm{R}}^n\mathrm{\hookrightarrow }{\mathrm{C}}^n$to complexify $\mathrm{A}$. In this case thecomplement${\mathrm{C}}^n\mathrm{\setminus }{\mathrm{\bigcup }}_{\mathrm{H}\mathrm{\in }\mathrm{A}}\mathrm{H}$usually has nontrivial fundamental group.