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Parameswaran Sankaran

Publications and source records attributed to Parameswaran Sankaran.

At least 19 recordsLinked to original sources
arXiv2026

On the fundamental groups of perforated surfaces

A perforated surface is the complement $\mathring\Sigma:=\Sigma\setminus A$ of a countable dense subset $A$ in a connected paracompact surface $\Sigma$. It is known that the topological type of $\Sigma\setminus A$ is independent of the choice of $A$. Any perforated surface is one-dimensional, connected, locally path connected, and is not semi-locally simply connected at any of its points. In this paper we obtain a classification theorem for perforated surfaces, using the classification theorem for surfaces. We show that any connected covering of a perforated surface $\mathring \Sigma$ arises from a covering of a surface $\Sigma'$ such that $\mathring\Sigma\cong \mathring\Sigma'$. We show that the fundamental group of perforated surfaces are large. We also show that the fundamental groups of $\mathring \Sigma$, the Sierpi\'nski curve and the Menger curve are not Hopfian.

math.GT↗
arXiv2024

Cohomology of flag bundles over compact Hermitian locally symmetric spaces

Let $E\to B$ be a complex analytic fiber bundle with fiber $F$, a flag variety over a compact complex manifold $B$. We shall obtain a description of the cohomology of $E$ when $B=X_\Gamma:=\Gamma\backslash X, E=Y_\Gamma:=\Gamma\backslash Y$ and $F=K/H$, a flag variety, where $Y=G/H$ and $X=G/K$, a Hermitian globally symmetric space of non-compact type with $G$ being a real, connected, non-compact, semisimple linear Lie group with no compact factors and simply connected complexification, $K\subset G$, a maximal compact subgroup, $H=Z_K(S)$, the centralizer in $K$ of a toral subgroup $S\subseteq K$ containing $Z(K)$, the center of $K$ and $\Gamma\subset G$, a uniform and torsionless lattice in $G$. We also obtain a description of the Picard group of $Y_\Gamma$ and $X_{\Gamma}$, for which the complexification of $G$ need not be simply connected. Moreover when $G$ is simple, we obtain the values of $ q$ for which $H^{p,q}(X_\Gamma)$ vanishes when $p=0,1$. This extends the results of R. Parthasarathy from $1980$, who considered (partially) the case $p=0$.

math.DG↗
arXiv2024

Cohomology and K-theory of generalized Dold manifolds fibred by complex flag manifolds

Let $\nu=(n_1,\ldots, n_s), s\ge 2,$ be a sequence of positive integers and let $n=\sum_{1\le j\le s}n_j$. Let $\mathbb CG(\nu)=U(n)/(U(n_1)\times \cdots\times U(n_s))$ be the complex flag manifold. Denote by $P(m,\nu)=P(\mathbb S^m,\mathbb CG(\nu))$ the generalized Dold manifold $\mathbb S^m\times \mathbb CG(\nu)/\langle \theta\rangle $ where $\theta=\alpha\times \sigma$ with $\alpha:\mathbb S^m\to \mathbb S^m$ being the antipodal map and $\sigma:\mathbb CG(\nu)\to \mathbb CG(\nu)$, the complex conjugation. The manifold $P(m,\nu)$ has the structure of a smooth $\mathbb CG(\nu)$-bundle over the real projective space $\mathbb RP^m.$ We determine the additive structure of $H^*(P(m,\nu);R)$ when $R=\mathbb Z$ and its ring structure when $R$ is a commutative ring in which $2$ is invertible. As an application, we determine the additive structure of $K(P(m,\nu))$ almost completely and also obtain partial results on its ring structure. The results for the singular homology are obtained for generalized Dold spaces $P(S,X)=S\times X/\langle \theta\rangle$, where $\theta=\alpha\times \sigma$, $\alpha:S\to S$ is a fixed point free involution and $\sigma:X\to X$ is an involution with $\mathrm{Fix}(\sigma)\ne \emptyset,$ for a much wider class of spaces $S$ and $X$.

math.AT↗
arXiv2023

THE K-RING OF E_6/Spin(10)

Let $\mathrm E_6$ denote the simply-connected compact exceptional Lie group of rank 6. The Lie group $\mathrm Spin(10)$ naturally embeds in $\mathrm E_6$, corresponding to the inclusion of the Dynkin diagrams. We determine the K-ring of the coset space ${\mathrm E}_6/\mathrm Spin(10)$. We identify the class of the tangent bundle of ${\mathrm E}_6/\mathrm Spin(10)$ in $KO({\mathrm E}_6/\mathrm Spin(10))$. As an application we show that ${\mathrm E}_6/\mathrm Spin(10)$ can be immersed in the Euclidean space $\mathbb R^{53}$.

math.KT↗
arXiv2022

K-theory of real Grassmann manifolds

Let $G_{n,k}$ denote the real Grassmann manifold of $k$-dimensional vector subspaces of $\mathbb R^n$. Using the Hodgkin spectral sequence, we compute the complex $K$-ring of $G_{n,k}$, up to a small indeterminacy, for all values of $n,k$ where $2\le k\le n-2$. When $n\equiv 0\!\!\mod 4, k\equiv 1\!\!\mod 2$ our result is complete.

math.KT↗
arXiv2022

Picard groups of certain compact complex parallelizable manifolds and related spaces

Let $G$ be a complex simply connected semisimple Lie group and let $Γ$ be a torsionless uniform irreducible lattice in $G$. Then $Γ\backslash G$ is a compact complex non-Kähler manifold whose tangent bundle is holomorphically trivial. In this note we compute the Picard group of $Γ\backslash G$ when $\rank(G)\geq 3$. When $\rank(G)\lneq 3$, we determine the group $Pic^0(Γ\backslash G)\subset Pic(Γ\backslash G)$ of topologically trivial holomorphic line bundles. When $\rank(G)\ge 2$, we also show that $Pic^0(P_Γ)$ is isomorphic to $Pic^0(Y)$ where $P_Γ$ is a $Γ\backslash G$-bundle associated to a principal $G$-bundle over a compact connected complex manifold $Y$, and, when $\rank(G)\ge 3$, we show that $Pic(Y)\to Pic(P_Γ)$ is injective with finite cokernel.

math.DG↗
arXiv2022

Twisted conjugacy in $SL_n$ and $GL_n$ over subrings of $\bar{\mathbb F}_p(t)$

Let $ϕ:G\to G$ be an automorphism of an infinite group $G$. One has an equivalence relation $\sim_ϕ$ on $G$ defined as $x\sim_ϕy$ if there exists a $z\in G$ such that $y=zxϕ(z^{-1})$. The equivalence classes are called $ϕ$-twisted conjugacy classes and the set $G/\!\!\sim_ϕ$ of equivalence classes is denoted $\mathcal R(ϕ)$. The cardinality $R(ϕ)$ of $\mathcal R(ϕ)$ is called the Reidemeister number of $ϕ$. We write $R(ϕ)=\infty$ when $\mathcal R(ϕ)$ is infinite. We say that $G$ has the $R_\infty$-{\it property} if $R(ϕ)=\infty$ for every automorphism $ϕ$ of $G$. We show that the groups $G=GL_n(R), SL_n(R)$ have the $R_\infty$-property for all $n\ge 3$ when $ F[t]\subset R\subsetneq F(t)$ where $F$ is a subfield of $\bar{\mathbb F}_p$. When $n\ge 4$, we show that any subgroup $H\subset GL_n(R)$ that contains $SL_n(R)$ also has the $R_\infty$-property.

math.GR↗
arXiv2022

K-theory of Springer varieties

The aim of this paper is to describe the topological $K$-ring, in terms of generators and relations, of a Springer variety $\mathcal{F}_{\lambda}$ of type $A$ associated to a nilpotent operator having Jordan canonical form whose block sizes form a weakly decreasing sequence $\lambda=(\lambda_1,\ldots, \lambda_l)$. Our description parallels the description of the integral cohomology ring of $\mathcal{F}_{\lambda}$ due to Tanisaki and also the equivariant analogue due to Abe and Horiguchi.

math.AT↗
arXiv2021

Cohomology of generalized Dold spaces

Let $(X,J) $ be an almost complex manifold with a (smooth) involution $σ:X\to X$ such that fix($σ$) is non-empty. Assume that $σ$ is a complex conjugation, i.e, the differential of $σ$ anti-commutes with $J$. The space $P(m,X):=\mathbb{S}^m\times X/\!\sim$ where $(v,x)\sim (-v,σ(x))$ was referred to as a generalized Dold manifold. The above definition admits an obvious generalization to a much wider class of spaces where $X, S$ are arbitrary topological spaces. The resulting space $P(S,X)$ will be called a generalized Dold space. When $S$ and $X$ are CW complexes satisfying certain natural requirements, we obtain a CW-structure on $P(S,X)$. Under certain further hypotheses, we determine the mod $2$ cohomology groups of $P(S,X)$. We determine the $\mathbb Z_2$-cohomology algebra when $X$ is (i) a torus manifold whose torus orbit space is a homology polytope, (ii) a complex flag manifold. One of the main tools is the Stiefel-Whitney class formula for vector bundles over $P(S,X)$ associated to $σ$-conjugate complex bundles over $X$ when the $S$ is a paracompact Hausdorff topological space, extending the validity of the formula, obtained earlier by Nath and Sankaran, in the case of generalized Dold manifolds.

math.AT↗
arXiv2021

Twisted conjugacy and commensurability invariance

A group $G$ is said to have property $R_{\infty}$ if for every automorphism $φ\in {\rm Aut}(G)$, the cardinality of the set of $φ$-twisted conjugacy classes is infinite. Many classes of groups are known to have such property. However, very few examples are known for which $R_{\infty}$ is {\it geometric}, i.e., if $G$ has property $R_{\infty}$ then any group quasi-isometric to $G$ also has property $R_{\infty}$. In this paper, we give examples of groups and conditions under which $R_{\infty}$ is preserved under commensurability. The main tool is to employ the Bieri-Neumann-Strebel invariant.

math.GR↗
arXiv2020

A note on the equivariant cobordism of generalized Dold manifolds

Let $(X,J) $ be an almost complex manifold with a (smooth) involution $σ:X\to X$ such that $Fix(σ)\neq \emptyset$. Assume that $σ$ is a complex conjugation, i.e, the differential of $σ$ anti-commutes with $J$. The space $P(m,X):=\mathbb{S}^m\times X/\!\sim$ where $(v,x)\sim (-v,σ(x))$ is known as a generalized Dold manifold. Suppose that a group $G\cong \mathbb Z_2^s$ acts smoothly on $X$ such that $g\circ σ=σ\circ g$ for all $g\in G$. Using the action of the diagonal subgroup $D=O(1)^{m+1}\subset O(m+1)$ on the sphere $\mathbb S^{m}$ for which there are only finitely many pairs of antipodal points that are stablized by $D$, we obtain an action of $\mathcal G=D\times G$ on $\mathbb S^m\times X$, which descends to a (smooth) action of $\mathcal G$ on $P(m,X)$. When the stationary point set $X^G$ for the $G$ action on $X$ is finite, the same also holds for the $\mathcal G$ action on $P(m,X)$. The main result of this note is that the equivariant cobordism class $[P(m,X),\mathcal G]$ vanishes if and only if $[X,G]$ vanishes. We illustrate this result in the case when $X$ is the complex flag manifold, $σ$ is the natural complex conjugation and $G\cong (\mathbb Z_2)^n$ is contained in the diagonal subgroup of $U(n)$.

math.AT↗
arXiv2020

Twisted conjugacy in free products

Let $ϕ:G\to G$ be an automorphism of a group which is a free-product of finitely many groups each of which is freely indecomposable and two of the factors contain proper finite index characteristic subgroups. We show that $G$ has infinitely many $ϕ$-twisted conjugacy classes. As an application, we show that if $G$ is the fundamental group of a three-manifold that is not irreducible, then $G$ has property $R_\infty$, that is, there are infinitely many $ϕ$-twisted conjugacy classes in $G$ for every automorphism $ϕ$ of $G$.

math.GR↗
arXiv2019

On generalized Dold manifolds

Let $X$ be a smooth manifold with a (smooth) involution $σ:X\to X$ such that $Fix(σ)\ne \emptyset$. We call the space $P(m,X):=\mathbb{S}^m\times X/\!\sim$ where $(v,x)\sim (-v,σ(x))$ a generalized Dold manifold. When $X$ is an almost complex manifold and the differential $Tσ: TX\to TX$ is conjugate complex linear on each fibre, we obtain a formula for the Stiefel-Whitney polynomial of $P(m,X)$ when $H^1(X;\mathbb{Z}_2)=0$. We obtain results on stable parallelizability of $P(m,X)$ and a very general criterion for the (non) vanishing of the unoriented cobordism class $[P(m,X)]$ in terms of the corresponding properties for $X$. These results are applied to the case when $X$ is a complex flag manifold.

math.AT↗
arXiv2018

The vector field problem for homogeneous spaces

The vector field problem is an important and classical problem in differential topology. In this survey we shall consider the vector field problem focusing mainly on the class of compact homogeneous spaces.

math.AT↗
arXiv2017

Geometric cycles in compact locally Hermitian symmetric spaces and automorphic representations

Let $G$ be a linear connected non-compact real simple Lie group and let $K\subset G$ be a maximal compact subgroup of $G$. Suppose that the centre of $K$ isomorphic to $\mathbb{S}^1$ so that $G/K$ is a global Hermitian symmetric space. Let $θ$ be the Cartan involution of $G$ that fixes $K$. Let $Λ$ be a uniform lattice in $G$ such that $θ(Λ)=Λ.$ Suppose that $G$ is one of the groups $SU(p,q), p<q-1, q\ge 5, SO_0(2,q)$, $Sp(n,\mathbb{R}), n\ne 4, SO^*(2n), n\ge 9.$ Then there exists a unique irreducible unitary representation $\mathcal{A}_\mathfrak{q}$ associated to a proper $θ$-stable parabolic subalgebra $\mathfrak{q}$ with $R_+(\mathfrak{q})=R_-(\mathfrak{q})$ such that if $H^{s,s}(\mathfrak{g},K;A_{\mathfrak{q}',K})\ne 0$ for some $0<s\le R_+(\mathfrak{q})$, then $\mathcal{A}_{\mathfrak{q}'}$ is unitarily equivalent to either the trivial representation or to $ \mathcal{A}_{\mathfrak{q}}$. As a consequence, under suitable hypotheses on $Λ,$ we show that the multiplicity of $\mathcal{A}_\mathfrak{q}$ occurring in $L^2(Γ\backslash G)$ is positive for {\it any} torsionless lattice $Γ\subset G$ commensurable with $Λ$.

math.RT↗
arXiv2016

Non-vanishing cohomology classes in uniform lattices of $\text{SO}(n,\mathbb{H})$ and automorphic representations

Let $X$ denote the non-compact globally Hermitian symmetric space of type $DIII$, namely, $\text{SO}(n,\mathbb{H})/\text{U}(n)$. Let $Λ$ be a uniform torsionless lattice in $\text{SO}(n,\mathbb{H})$. In this note we construct certain complex analytic submanifolds in the locally symmetric space $X_Γ:=Γ\backslash \text{SO}(n,\mathbb{H})/\text{U}(n)$ for certain finite index sub lattices $Γ\subset Λ$ and show that their dual cohomology classes in $H^*(X_Γ;\mathbb{C})$ are not in the image of the Matsushima homomorphism $H^*(X_u; \mathbb{C})\to H^*(X_Γ;\mathbb{C})$, where $X_u=\text{SO}(2n)/\text{U}(n)$ is the compact dual of $X$. These submanifold arise as sub-locally symmetric spaces which are totally geodesic, and, when $Λ$ satisfies certain additional conditions, they are non-vanishing `special cycles'. Using the fact that $X_Λ$ is a Kähler manifold, we deduce the occurrence in $L^2(Λ\backslash \text{SO}(n,\mathbb{H})$ of a certain irreducible representation $(\mathcal{A}_\mathfrak{q}, A_\mathfrak{q})$ with non-zero multiplicity when $n\ge 9$. The representation $\mathcal{A}_\mathfrak{q}$ is associated to a certain $θ$-stable parabolic subalgebra $\mathfrak{q}$ of $\mathfrak{g}_0:=\mathfrak{so}(n,\mathbb{H})$. Denoting the smooth $\text{U}(n)$-finite vectors of $A_{\mathfrak{q}}$ by $A_{\mathfrak{q},\text{U}(n)}$, the representation $\mathcal{A}_\mathfrak{q}$ is characterised by the property that $H^{p,p}(\mathfrak{g}_0\otimes\mathbb{C},\text{U}(n); A_{\mathfrak{q},\text{U}(n)})\cong H^{p-n+2,p-n+2}(\text{SO}(2n-2)/\text{U}(n-1);\mathbb{C}),~p\ge 0$, for $n\ge 9$.

math.RT↗