  Physics in the neighbourhood of a space-time metric singularity is described
by a world-sheet topological gauge field theory which can be represented as a
twisted $N=2$ superconformal Wess-Zumino model with a $W_{1+\infty} \otimes
W_{1+\infty} $ bosonic symmetry. The measurable $W$-hair associated with the
singularity is associated with Wilson loop integrals around gauge defects. The
breaking of $W_{1+\infty}$ $\otimes $ $W_{1+\infty}$ $\rightarrow $
$W_{1+\infty}$ is associated with expectation values for open Wilson lines that
make the metric non-singular away from the singularity. This symmetry breaking
is accompanied by massless discrete `tachyon' states that appear as leg poles
in $S$-matrix elements. The triviality of the $S$-matrix in the high-energy
limit of the $c=1$ string model, after renormalisation by the leg pole factors,
is due to the restoration of double $W$-symmetry at the singularity.
