  The standard model of cosmology with postulated dark energy and dark matter
sources may be considered as a fairly successful fitting model to observational
data. However, this model leaves the question of the physical origin of these
dark components open. Fully relativistic contributions that act like dark
energy on large scales and like dark matter on smaller scales can be found
through generalization of the standard model by spatially averaging the
inhomogeneous Universe within general relativity. The spatially averaged 3+1
Einstein equations are effective balance equations that need a closure
condition. Heading for closure we here explore topological constraints. Results
are straightforwardly obtained for averaged 2+1 model universes. For the
relevant 3+1 case, we employ a method based on the Gauss-Bonnet-Chern theorem
generalized to Lorentzian spacetimes and implement a sandwich approach to
obtain spatial average properties. The 3+1 topological approach supplies us
with a new equation linking evolution of scalar invariants of the expansion
tensor to the norm of the Weyl tensor. From this we derive general evolution
equations for averaged scalar curvature and kinematical backreaction, and we
discuss related evolution equations on this level of the hierarchy of averaged
equations. We also discuss the relation between topological properties of
cosmological manifolds and dynamical topology change, e.g. as resulting from
the formation of black holes.
