On products in a real moment-angle manifold

Abstract

In this paper we give a necessary and sufficient condition for a (real) moment-angle complex to be a topological manifold. The cup and cap products in a real moment-angle manifold are studied: the Poincaré duality via cap products is equivalent to the Alexander duality of the defining complex K . Consequently, the cohomology ring (with coefficients integers) of a polyhedral product by pairs of disks and their bounding spheres is isomorphic to that of a differential graded algebra associated to K , and the dimensions of the disks.2010 Mathematics Subject Classification. Primary 55N45; Secondary 57Q15, 32S22.

A spectral sequence for polyhedral products

“…The following is proven in [6]. The left lexicographical ordering of the simplices of K induce a filtration on C K = I Z{y σ |σ ∈ K I }.…”

“…Next observe that the shuffle map on t i and s i is exactly the map induced by the shuffle map for H * (Σ|N J 1 |) ⊗ H * (Σ|N J 2 |) → H * (Σ|N J=J 1 ∪J 2 |) Which by example 6.6, is the * −product. Using work of Cai [6] we will give a chain level formula for the * −product in section 7.…”

Polyhedral products and commutator subgroups of right-angled Artin and Coxeter groups

“…The space R K was introduced and studied in the works of Davis [Da1] and Davis-Januszkiewicz [DJ], although their construction was different. When |K| is homeomorphic to a sphere, R K is a topological manifold (this follows from the results of [Da1], see also [BP2] and [Ca,Theorem 2.3]). Furthermore, the manifold R K has a smooth structure when |K| is the boundary of a convex polytope.…”

Multiplicative structure of the cohomology ring of real toric spaces

“…In this section, we study the cohomology ring of a real moment-angle complex using a natural CW structure of the cube (D 1 ) m . We basically follow the arguments of [4] and [10], but with the basis (2.2) which causes huge difference as we can see in Section 3. We will use the notation C * (X) and C * (X) for the simplicial or cellular (co)chain complex of X when X is a simplicial complex or a CW complex respectively.…”

“…(1) For Z K , (2.9) holds for an arbitrary coefficient ring R. Indeed, we could choose the basis (2.3) to obtain the result of [4] for H * (RZ K ; R) for arbitrary coefficient. (2) The difference of (2.4) and (2.8) yields significant contrast of the rings H * (RZ K ; R) and H * (Z K ; R).…”