  Curvature and torsion are the two tensors characterizing a general Riemannian
spacetime. In Einstein's general theory of gravitation, with torsion postulated
to vanish and the affine connection identified to the Christoffel symbol, only
the curvature tensor plays the central role. For such a purely metric geometry,
two well-known topological invariants, namely the Euler class and the
Pontryagin class, are useful in characterizing the topological properties of
the spacetime. From a gauge theory point of view, and especially in the
presence of spin, torsion naturally comes into play, and the underlying
spacetime is no longer purely metric. We describe a torsional topological
invariant, discovered in 1982, that has now found increasing usefulness in
recent developments.

General Relativity and Quantum Cosmology