Soumen Sarkar

The simplicial wedge construction on simplicial complexes and simple polytopes has been used by a variety of authors to study toric and related spaces, including non-singular toric varieties, toric manifolds, intersections of quadrics and more generally, polyhedral products. In this paper we extend the analysis to include toric orbifolds. Our main results yield infinite families of toric orbifolds, derived from a given one, whose integral cohomology is free of torsion and is concentrated in even degrees, a property which might be termed integrally equivariantly formal. In all cases, it is possible to give a description of the cohomology ring and to relate it to the cohomology of the original orbifold.

Torus orbifolds are topological generalizations of symplectic toric orbifolds. The authours give a construction of smooth orbifolds with torus actions whose boundary is a disjoint union of torus orbifolds using a toric topological method. As a result, they show that any orientable locally standard torus orbifold is equivariantly cobordant to some copies of orbifold complex projective spaces. They also discuss some further equivariant cobordism results including the cases when torus orbifolds are actually torus manifolds.

In this paper, we study some properties of retraction sequences, blowups of polytopes and their interrelations. We introduce the blowups of quasitoric orbifolds using the combinatorial data associated with its orbit spaces. We prove that neither new singularity nor new torsion arises in the integral cohomologies of certain blowups of a quasitoric orbifold. Moreover, we show that if a quasitoric orbifold has no p-torsion in its cohomology then the cohomology of its blowup along a fixed point has no p-torsion. As a consequence, we construct infinitely many integrally equivariantly formal quasitoric orbifolds from a given one. We also investigate the torsions in the cohomologies of blowups of a class of simplicial toric varieties.

We introduce the category of locally k –standard T –manifolds, which includes well-known classes of manifolds such as toric and quasitoric manifolds, good contact toric manifolds and moment–angle manifolds. They are smooth manifolds with well-behaved actions of tori. We study their topological properties, such as fundamental groups and equivariant cohomology algebras. Then we discuss when the torus equivariant cohomology algebra distinguishes them up to weakly equivariant homeomorphism.

In this paper, we study three relative LS categories of a map and study some of their properties. Then we introduce the ‘higher topological complexity’ and ‘weak higher topological complexity’ of a map. Each of them are homotopy invariants. We discuss some lower and upper bounds of these invariants and compare them with previously known ‘topological complexities’ of a map.

The CW-complex structure of certain spaces, such as effective orbifolds, can be too complicated for computational purposes. In this paper we develop the concept of $\mathbf{q}$-CW complex structure on an orbifold, to detect torsion in its integral cohomology. The main result can be applied to well-known classes of orbifolds or algebraic varieties having orbifold singularities, such as toric orbifolds, simplicial toric varieties, torus orbifolds, and weighted Grassmannians.

We study the GKM theory for a equivariant stratified space having orbifold structures in its successive quotients. Then, we introduce the notion of an almost simple polytope, as well as a divisive toric variety generalizing the concept of a divisive weighted projective space. We employ the GKM theory to compute the generalized equivariant cohomology theories of toric varieties associated to almost simple polytopes and divisive toric varieties.

In this paper, we study the relation between the Lusternik– Schnirelmann category and the topological complexity of two closed orien-ted manifolds connected by a degree one map.

We introduce the concepts of orbit class, orbit diagram and orbital spaces, and study their properties. Then we apply the results on orbit classes to study equivariant and invariant topological complexity, as well as to give some lower and upper bounds for the equivariant LS-category and A-G-LS-category.

The category of torus orbifolds is a generalization of the category of toric orbifolds which contains projective toric varieties associated to complete simplicial fans. We introduce the concept of “divisive” torus orbifolds following divisive weighted projective spaces. The divisive condition may ensure an invariant cell structure on a locally standard torus orbifold. We give a combinatorial description of equivariant K-theory, equivariant cobordism theory and equivariant cohomology theory of divisive torus orbifolds. In particular, we get a combinatorial description of these generalize cohomology theories for torus manifolds over acyclic polytopes.