  In this paper, we study the well-know $g$-conjecture for rational homology
spheres in a topological way. To do this, we construct a class of topological
spaces with torus actions, which can be viewed as topological generalizations
of toric varieties. Along this way we prove that after doing stellar
subdivision operations at some middle dimensional faces of an arbitrary
rational homology sphere, the $g$-conjecture is valid. Furthermore, we give
topological proofs of several fundamental algebraic results about Buchsbaum
complexes and simplicial manifolds. In this process, we also get a few
interesting results in toric topology.
