We generalize the notion of moment-angle manifold over a simple convex polytope to
an arbitrary nice manifold with corners. For a nice manifold with corners
,
we first compute the stable decomposition of the moment-angle manifold
via a construction called
rim-cubicalization of
.
From this, we derive a formula to compute the integral cohomology group of
via the
strata of
.
This generalizes the Hochster’s formula for the moment-angle manifold over a simple
convex polytope. Moreover, we obtain a description of the integral cohomology ring
of
using the idea of partial diagonal maps. In addition, we define the
notion of polyhedral product of a sequence of based CW–complexes over
and obtain similar results for these spaces as we do for
.
Using this general construction, we can compute the equivariant cohomology ring of
with
respect to its canonical torus action from the Davis–Januszkiewicz space of
. The
result leads to the definition of a new notion called the topological face ring of
, which
generalizes the notion of face ring of a simple polytope. Moreover, the topological face
ring of
computes the equivariant cohomology of all locally standard torus actions with
as the orbit space when the corresponding principal torus bundle over
is trivial.
Meanwhile, we obtain some parallel results for the real moment-angle manifold
over
as
well.
Keywords
moment-angle manifold, topological face ring, manifold with
corners, equivariant cohomology, Davis–Januszkiewicz space
