  The singularity present in cosmological instantons of the Hawking-Turok type
is resolved by a conformal transformation, where the conformal factor has a
linear zero of codimension one. We show that if the underlying regular manifold
is taken to have the topology of $RP^4$, and the conformal factor is taken to
be a twisted field so that the zero is enforced, then one obtains a
one-parameter family of solutions of the classical field equations, where the
minimal action solution has the conformal zero located on a minimal volume
noncontractible $RP^3$ submanifold. For instantons with two singularities, the
corresponding topology is that of a cylinder $S^3\times [0,1]$ with D=4
analogues of `cross-caps' at each of the endpoints.
