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.
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