  We consider a Poisson point process on the space of lines in R^d, where a
multiplicative factor u>0 of the intensity measure determines the density of
lines. Each line in the process is taken as the axis of a bi-infinite cylinder
of radius 1. We investigate percolative properties of the vacant set, defined
as the subset of R^d that is not covered by any such cylinder. We show that in
dimensions d >= 4, there is a critical value u_*(d) \in (0,\infty), such that
with probability 1, the vacant set has an unbounded component if u<u_*(d), and
only bounded components if u>u_*(d). For d=3, we prove that the vacant set does
not percolate for large u and that the vacant set intersected with a
two-dimensional subspace of R^d does not even percolate for small u>0.
