Integral cohomology rings of weighted Grassmann orbifolds and rigidity properties

Abstract

In this paper, we introduce `Pl\"{u}cker weight vector' and establish the definition of a weighted Grassmann orbifold ${\rm Gr}_{\bf b}(k,n)$, corresponding to a Pl\"{u}cker weight vector `${\bf b}$'. We achieve an explicit classification of weighted Grassmann orbifolds up to certain homeomorphism in terms of the Pl\"{u}cker weight vectors. We study the integral cohomology of ${\rm Gr}_{\bf b}(k,n)$ and provide some sufficient conditions such that the integral cohomology of ${\rm Gr}_{\bf b}(k,n)$ has no torsion. We explicitly describe the formula of the equivariant structure constants with respect to the equivariant Schubert basis in equivariant cohomology ring of divisive weighted Grassmann orbifolds with integer coefficients. Eminently, we compute the integral cohomology rings of divisive weighted Grassmann orbifolds explicitly.