  We present a self-gravitating, analytic and globally regular Skyrmion
solution of the Einstein-Skyrme system with winding number w = 1, in presence
of a cosmological constant. The static spacetime metric is the direct product
RxS3 and the Skyrmion is the self-gravitating generalization of the static
hedgehog solution of Manton and Ruback with unit topological charge. This
solution can be promoted to a dynamical one in which the spacetime is a
cosmology of the Bianchi type-IX with time-dependent scale and squashing
coefficients. Remarkably, the Skyrme equations are still identically satisfied
for all values of these parameters. Thus, the complete set of field equations
for the Einstein-Skyrme-Lambda system in the topological sector reduces to a
pair of coupled, autonomous, nonlinear differential equations for the scale
factor and a squashing coefficient. These equations admit analytic bouncing
cosmological solutions in which the universe contracts to a minimum
non-vanishing size, and then expands. A non-trivial byproduct of this solution
is that a minor modification of the construction gives rise to a family of
stationary, regular configurations in General Relativity with negative
cosmological constant supported by an SU(2) nonlinear sigma model. These
solutions represent traversable AdS wormholes with NUT parameter in which the
only "exotic matter" required for their construction is a negative cosmological
constant.
