arXiv subjects

Soumen Sarkar

Publications and source records attributed to Soumen Sarkar.

54 records · Page 3Linked to original sources

arXiv2019

Equivariant cobordism of torus orbifolds

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

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arXiv2018

Cohomology Rings of a Class of Torus Manifolds

Torus manifolds are topological generalization of smooth projective toric manifolds. We compute the rational cohomology ring of a class of smooth locally standard torus manifolds whose orbit space is a connected sum of simple polytopes.

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arXiv2018

Higher equivariant and invariant topological complexity

In this paper we introduce the concepts of higher equivariant and invariant topological complexity; and study their properties. Then we compare them with equivariant LS-category. We give lower and upper bounds for these new invariants. We compute some of these invariants for moment angle complexes.

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arXiv2018

Infinite Families of Equivariantly Formal Toric Orbifolds

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 \emph{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.

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arXiv2017

On integral cohomology of certain orbifolds

The CW structure of certain spaces, such as effective orbifolds, can be too complicated for computational purposes. In this paper we use 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.

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arXiv2016

On the Integral Cohomology Ring of Toric Orbifolds and Singular Toric Varieties

We examine the integral cohomology rings of certain families of $2n$-dimensional orbifolds $X$ that are equipped with a well-behaved action of the $n$-dimensional real torus. These orbifolds arise from two distinct but closely related combinatorial sources, namely from characteristic pairs $(Q,λ)$, where $Q$ is a simple convex $n$-polytope and $λ$ a labelling of its facets, and from $n$-dimensional fans $Σ$. In the literature, they are referred as toric orbifolds and singular toric varieties respectively. Our first main result provides combinatorial conditions on $(Q,λ)$ or on $Σ$ which ensure that the integral cohomology groups $H^{\ast}(X)$ of the associated orbifolds are concentrated in even degrees. Our second main result assumes these condition to be true, and expresses the graded ring $H^*(X)$ as a quotient of an algebra of polynomials that satisfy an integrality condition arising from the underlying combinatorial data. Also, we compute several examples.

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arXiv2016

Some Aspects of Equivariant LS-category

We study the lower bounds and upper bounds for LS-category and equivariant LS-category. In particular we compute both invariants for torus manifolds. There are some examples to show the sharpness of conditions in the theorems. Moreover the equivariant LS-category of the product space is discussed and counterexamples of some previous results are given.

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arXiv2016

Orientability and equivariant oriented cobordism of 2-torus manifolds

We give a necessary and sufficient condition for the orientability of a locally standard 2-torus manifold with a fixed point which generalizes previous results of Nakayama-Nishimura in 2005 and Soprunova-Sottile in 2013. We construct manifolds with boundary where the boundary is a disjoint union of locally standard 2-torus manifolds. We discuss equivariant oriented cobordism class of locally standard 2-torus manifolds.

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arXiv2016

A new construction of lens spaces

Let $T^n$ be the real $n$-torus group. We give a new definition of lens spaces and study the diffeomorphic classification of lens spaces. We show that any $3$-dimensional lens space $L(p; q)$ is $T^2$-equivariantly cobordant to zero. We also give some sufficient conditions for higher dimensional lens spaces $L(p; q_1, \ldots, q_n)$ to be $T^{n+1}$-equivariantly cobordant to zero. In 2005, B. Hanke showed that complex equivariant cobordism class of a lens space is trivial. Nevertheless, our proofs are constructive using toric topological arguments.

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arXiv2016

Complex Cobordism of quasitoric orbifolds

We construct orbifolds with quasitoric boundary and show that they have stable almost complex structure. We show that a quasitoric orbifold is complex cobordant to finite disjoint copies of complex orbifold projective spaces. Finally some computations in the complex cobordism ring for manifolds are given.

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arXiv2015

Equilibrium triangulations of some quasitoric 4-manifolds

Quasitoric manifolds, introduced by M. Davis and T. Januskiewicz in 1991, are topological generalizations of smooth complex projective spaces. In 1992, Banchoff and Kühnel constructed a 10-vertex equilibrium triangulations of $\CP^2$. We generalize this construction for quasitoric manifolds and construct some equilibrium triangulations of $4$-dimensional quasitoric manifolds. In some cases, our constructions give vertex minimal equilibrium triangulations.

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arXiv2015

Triangulations of $\mathbb{RP}^n$ with few vertices

P. Arnoux and A. Marin showed that any triangulation of $\mathbb{RP}^n$ contains more than $\frac{(n+1)(n+2)}{2}$ vertices if $n \geq 3$. We construct some natural triangulation of $\mathbb{RP}^n$ with $\frac{n(n+5)}{2}-1$ vertices for all $n \geq 3$. Previously, it was known that $\mathbb{RP}^n$ has $\mathbb{Z}_2^n$-equivariant triangulation with $n(n+1)$ vertices for $n \geq 6$.

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arXiv2014

Equilibrium and equivariant triangulations of some small covers with minimum number of vertices

Small covers were introduced by Davis and Januszkiewicz in 1991. We introduce the notion of equilibrium triangulations for small covers. We study equilibrium and vertex minimal $\mathbb{Z}_2^2$-equivariant triangulations of $2$-dimensional small covers. We discuss vertex minimal equilibrium triangulations of $\mathbb{RP}^3 \# \mathbb{RP}^3$, $S^1 \times \mathbb{RP}^2$ and a nontrivial $S^1$ bundle over $\mathbb{RP}^2$. We construct some nice equilibrium triangulations of the real projective space $\mathbb{RP}^n$ with $2^n +n+1$ vertices. The main tool is the theory of small covers.

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arXiv2014

A class of torus manifolds with nonconvex orbit space

We study a class of smooth torus manifolds whose orbit space has the combinatorial structure of a simple polytope with holes. We construct moment angle manifolds for such polytopes with holes and use them to prove that the associated torus manifolds admit stable almost complex structure. We give a combinatorial formula for the Hirzebruch $χ_y$ genus of these torus manifolds. We show that they have (invariant) almost complex structure if they admit positive omniorientation. We give examples of almost complex manifolds that do not admit a complex structure. When the dimension is four, we calculate the homology groups and describe a method for computing the cohomology ring.

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arXiv2012

$\TT^2$-cobordism of Quasitoric 4-Manifolds

We show the $\TT^2$-cobordism group of the category of 4-dimensional quasitoric manifolds is generated by the $\TT^2$-cobordism classes of $\CP^2$. We construct nice oriented $\TT^2$ manifolds with boundary where the boundary is the Hirzebruch surfaces. The main tool is the theory of quasitoric manifolds.

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arXiv2010

Some small orbifolds over polytopes

We introduce some compact orbifolds on which there is a certain finite group action having a simple convex polytope as the orbit space. We compute the orbifold fundamental group and homology groups of these orbifolds. We calculate the cohomology rings of these orbifolds when the dimension of the orbifold is even. These orbifolds are intimately related to the notion of small cover.

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arXiv2010

Oriented Cobordism of $\CP^{2k+1}$

We give a new construction of oriented manifolds having the boundary $\CC P^{2k+1}$ for each $k \geq 0$. The main tool is the theory of quasitoric manifolds.

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arXiv2009

On Quasitoric Orbifolds

Quasitoric spaces were introduced by Davis and Januskiewicz in their 1991 Duke paper. There they extensively studied topological invariants of quasitoric manifolds. These manifolds are generalizations or topological counterparts of nonsingular projective toric varieties. In this article we study structures and invariants of quasitoric orbifolds. In particular, we discuss equivalent definitions and determine the orbifold fundamental group, rational homology groups and cohomology ring of a quasitoric orbifold. We determine whether any quasitoric orbifold can be the quotient of a smooth manifold by a finite group action or not. We prove existence of stable almost complex structure and describe the Chen-Ruan cohomology groups of an almost complex quasitoric orbifold.

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