  The topological containment problem is known to be polynomial-time solvable
for any fixed pattern graph $H$, but good characterisations have been found for
only a handful of non-trivial pattern graphs. The complete graph on five
vertices, $K_5$, is one pattern graph for which a characterisation has not been
found. The discovery of such a characterisation would be of particular
interest, due to the Haj\'os Conjecture. One step towards this may be to find a
good characterisation of graphs that do not topologically contain the simpler
pattern graph $K_5^-$, obtained by removing a single edge from $K_5$.
  This paper makes progress towards achieving this, by showing that every
4-connected graph must contain a $K_5^-$-subdivision.
