  Exotic smooth manifolds, ${\bf R^2\times_\Theta S^2}$, are constructed and
discussed as possible space-time models supporting the usual Kruskal
presentation of the vacuum Schwarzschild metric locally, but {\em not
globally}. While having the same topology as the standard Kruskal model, none
of these manifolds is diffeomorphic to standard Kruskal, although under certain
conditions some global smooth Lorentz-signature metric can be continued from
the local Kruskal form. Consequently, it can be conjectured that such manifolds
represent an infinity of physically inequivalent (non-diffeomorphic) space-time
models for black holes. However, at present nothing definitive can be said
about the continued satisfaction of the Einstein equations. This problem is
also discussed in the original Schwarzschild $(t,r)$ coordinates for which the
exotic region is contained in a world tube along the time-axis, so that the
manifold is spatially, but not temporally, asymptotically standard. In this
form, it is tempting to speculate that the confined exotic region might serve
as a source for some exterior solution. Certain aspects of the Cauchy problem
are also discussed in terms of ${\bf R^4_\\Theta}$\ models which are
``half-standard'', say for all $t<0,$ but for which $t$ cannot be globally
smooth.
