  We find an exact time-dependent instanton solution on a vacuum Kerr-like
warped spacetime in conformal dilaton gravity. Remarkably, the metric solution
results from a first-order PDE, allowing the connection with self-duality. The
antipodal boundary condition on the hypersurface of a Klein bottle $\sim
\mathbb{C}^1\times\mathbb{C}^1$ is applied to describe the Hawking particles.
We used the Hopf fibration to get $S^2$ as the black hole horizon, where the
centrix is not in a torus but in the Klein bottle. The twist fits very well
with the antipodal identification of the point on the horizon. No "cut and
past" is necessary, so the Hawing particles remain pure without instantaneous
information transport. A local observer passing the horizon will not notice a
central singularity in suitable coordinates. The black hole paradoxes are also
revisited in our new black hole model. A connection is made with the geomeric
quantization of $\mathbb{C}^1\times\mathbb{C}^1\sim S^3$, by considering the
symplectic 2-form. The model can be easily extended to the non-vacuum situation
by including a scalar field. Both the dilaton and the scalar field can be
treated as quantum fields when approaching the Planck area.
