On the cohomology and their torsion of real toric objects

Abstract

Abstract. In this paper, we do the two things.(1) We present a formula to compute the rational cohomology ring of a real topological toric manifold, and thus that of a small cover or a real toric manifold, which implies the formula of Suciu and Trevisan. Furthermore, the formula also works for other coefficient Zq = Z/qZ, where q is a positive odd integer. (2) We construct infinitely many real toric manifolds and small covers whose integral cohomology have a q-torsion for any positive odd integer q.

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.…”

“…This is one of the key properties of the basis change (2.2). It leads the simpler and improved cohomology ring formula of M R (K, Λ) as in Theorem 3.4 below rather than one given in [10].…”

“…It, thus, is reasonable to expect that for α ∈ H p−1 (K ω 1 ) and β ∈ H q−1 (K ω 2 ), the cup product α ⌣ β is in H p+q−1 (K ω 1 +ω 2 ). However, the formula given in [10,Theorem 4.5] does not guarantee in transparent way that the natural group isomorphism such as in Theorem 1.1 satisfies the property.…”

“…[10, Theorem 4.6] Let M = M R (K, Λ) be a real toric space and R a commutative ring in which 2 is a unit. Then there is an R-linear isomorphism…”

A moment-angle manifold whose cohomology has torsion

“…(2) Similarly we make stellar subdivision at (3, 7) on L 0 at first, then make stellar subdivision at (1, 2, 3), the resulting set of missing faces of L ′ 2 is M ′ 2 = {(3, 7), (1,2,4,5,6,8,9), (1, 2, 3), (7, 10), (4,5,6,8,9,10)} . It is easy to see that two simplicial complexes K and K ′ on vertex set I are combinatorial equivalent if and only if their sets of missing faces M and M ′ are equivalent, i.e.…”