A torus manifold is an even-dimensional
manifold acted on by a half-dimensional torus with non-empty
fixed point set and some additional orientation data. It may
be considered as a far-reaching generalisation of toric
manifolds from algebraic geometry. The orbit space of
a torus manifold has a rich combinatorial structure, e.g.,
it is a manifold with corners provided that the action
is locally standard. Here we investigate relationships
between the cohomological properties of torus manifolds and
the combinatorics of their orbit quotients. We show that the
cohomology ring of a torus
manifold is generated by two-dimensional classes if and only
if the quotient is a homology polytope. In this case
we retrieve the familiar picture from toric geometry: the
equivariant cohomology is the face ring of the nerve
simplicial complex and the ordinary cohomology is obtained
by factoring out certain linear forms. In a more general situation,
we show that the odd-degree cohomology of a torus manifold
vanishes if and only if the orbit space is face-acyclic.
Although the cohomology is no longer generated in degree
two under these circumstances, the equivariant cohomology
is still isomorphic to the face ring of an appropriate simplicial
poset.