arXiv subjects

Ramesh Kasilingam

Publications and source records attributed to Ramesh Kasilingam.

15 recordsLinked to original sources

arXiv2026

Diffeomorphism Classification of Smooth Structures and Tangential Homotopy Types of $\mathbb{C}P^m$ for $5 \le m \le 8$

This paper provides a diffeomorphism classification of smooth manifolds homeomorphic to the complex projective space $\mathbb{C}P^m$ for $m \in \{5, 6, 7, 8\}$. The classification is obtained by computing the group of concordance classes of smooth structures on $\mathbb{C}P^m$ and determining the orbit space under the action induced by the group of self-homeomorphisms. Using these computations in conjunction with the tangential surgery exact sequence and techniques from stable homotopy theory, we determine the diffeomorphism classes of smooth manifolds within the tangential homotopy type of $\mathbb{C}P^m$ for $4 \le m \le 8$. We also investigate the relationship between these two classification problems by studying the natural map from the homeomorphism type to the tangential homotopy type. As a consequence, we prove that for $m = 4$, there exists a unique smooth manifold, up to diffeomorphism, that is tangentially homotopy equivalent to $\mathbb{C}P^4$ but not homeomorphic to it. Furthermore, for $m = 8$, there exist exactly two pairwise non-diffeomorphic smooth manifolds that are tangentially homotopy equivalent to $\mathbb{C}P^8$ but not homeomorphic to it.

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arXiv2025

Enumerating Smooth Structures on $\mathbb{C}P^3\times\mathbb{S}^k$

In this paper, we compute the concordance inertia group of the product $M \times \mathbb{S}^k$, where $M$ is a simply connected, closed, smooth 6-manifold, for $1 \leq k \leq 10$, using known low-dimensional computations of the stable homotopy groups of spheres. Specifically, for $M = \mathbb{C}P^3$, we determine the inertia group of $\mathbb{C}P^3 \times \mathbb{S}^k$ for $2 \leq k \leq 7, k \neq 6$, and establish a diffeomorphism classification of all smooth manifolds homeomorphic to $\mathbb{C}P^3 \times \mathbb{S}^k$ for $1 \leq k \leq 7$.

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arXiv2024

On the Classification of Planar-Rips complexes and their corresponding unit disk graphs

Given a metric space $(X,d)$, the Vietoris-Rips complex of $X$ at a scale of $r >0$ is a simplicial complex whose simplices are all those finite subsets of $X$ with diameter less than $r$. In this paper, we classify, up to simplicial isomorphism, all $n$-dimensional pseudomanifolds and weak-pseudomanifolds that can be realized as a Vietoris-Rips complex of planar point sets. We further classify two-dimensional, pure, and closed planar-Rips complexes up to homotopy. Additionally, we explore the hereditary properties and introduce the notion of obstructions in planar-Rips complexes. We also consolidate our findings to describe a class of unit disk graphs, having all maximal cliques of same cardinality. Several structural and geometric properties of planar-Rips complexes have also been derived.

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arXiv2024

Smooth Structures on $M\times\mathbb{S}^k$

This paper explores various differentiable structures on the product manifold $M \times \mathbb{S}^k$, where $M$ is either a 4-dimensional closed, oriented, smooth manifold or a simply connected 5-dimensional closed, smooth manifold. We identify the possible stable homotopy types of $M$ and use it to calculate the concordance inertia group and the concordance structure set of $M\times\mathbb{S}^k$ for $1\leq k\leq 10$. These calculations enable us to further classify all manifolds that are homeomorphic to $\mathbb{C}P^2\times\mathbb{S}^k$, up to diffeomorphism, for each $4\leq k\leq 6$.

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arXiv2023

Smooth structures on PL-manifolds of dimensions between 8 and 10

In this paper, we identify the concordance classes of smooth structures on $PL$-manifolds of dimension between $8$ and $10$ in terms of the cohomology and Steenrod operations. This leads to the computation of the homotopy inertia groups. Finally we discuss the special cases of Lens spaces and real projective spaces.

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arXiv2017

Inertia Groups and Smooth Structures on Quaternionic Projective Spaces

This paper deals with certain results on the number of smooth structures on quaternionic projective spaces, obtained through the computation of inertia group and its analogues, which in turn are computed using techniques from stable homotopy theory. We show that the concordance inertia group is trivial in dimension 20, but there are many examples in high dimensions where the concordance inertia group is non-trivial. We extend these to computations of concordance classes of smooth structures. These have applications to 3-sphere actions on homotopy spheres and tangential homotopy structures.

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arXiv2017

Homotopy Inertia Groups and Tangential Structures

We show that if $M$ and $N$ have the same homotopy type of simply connected closed smooth $m$-manifolds such that the integral and mod-$2$ cohomologies of $M$ vanish in odd degrees, then their homotopy inertia groups are equal. Let $M^{2n}$ be a closed $(n-1)$-connected $2n$-dimensional smooth manifold. We show that, for $n=4$, the homotopy inertia group of $M^{2n}$ is trivial and if $n=8$ and $H^n(M^{2n};\mathbb{Z})\cong \mathbb{Z}$, the homotopy inertia group of $M^{2n}$ is also trivial. We further compute the group $\mathcal{C}(M^{2n})$ of concordance classes of smoothings of $M^{2n}$ for $n=8$. Finally, we show that if a smooth manifold $N$ is tangentially homotopy equivalent to $M^8$, then $N$ is diffeomorphic to the connected sum of $M^8$ and a homotopy $8$-sphere.

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arXiv2017

Smooth structures on $\mathbb{C}P^{m}$ for $5\leq m\leq 8$

We classify up to diffeomorphism all smooth manifolds homeomorphic to the complex projective m-space $\mathbb{C}P^{m}$ for $m = 5, 6, 7$ and $8$. As an application, for $m = 7$ and $8$, we compute the smooth tangential structure set of $\mathbb{C}P^{m}$ and obtain a bound on the number of smooth homotopy complex projective m-spaces with given Pontryagin classes up to orientation-preserving diffeomorphism. We also show that there exists a smooth manifold which is tangentially homotopy equivalent but not homeomorphic to $\mathbb{C}P^{8}$.

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arXiv2016

Smooth Structures on a Fake Real Projective Space

We show that the group of smooth homotopy $7$-spheres acts freely on the set of smooth manifold structures on a topological manifold $M$ which is homotopy equivalent to the real projective $7$-space. We classify, up to diffeomorphism, all closed manifolds homeomorphic to the real projective $7$-space. We also show that $M$ has, up to diffeomorphism, exactly $28$ distinct differentiable structures with the same underlying PL structure of $M$ and $56$ distinct differentiable structures with the same underlying topological structure of $M$.

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arXiv2015

Topological Rigidity Problems

We survey the recent results and current issues on the topological rigidity problem for closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. A number of open problems and conjectures are presented during the course of the discussion. We also review the status and applications of the Farrell-Jones Conjecture for algebraic $K$-and $L$-theory for a group ring $RG$ and coefficients in an additive category. These conjectures imply many other well-known and important conjectures. Examples are the Borel Conjecture about the topological rigidity of closed aspherical manifolds, the Novikov Conjecture about the homotopy invariance of higher signatures and the Conjecture for vanishing of the Whitehead group. We here present the status of the Borel, Novikov and vanishing of the Whitehead group Conjectures.

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arXiv2015

Inertia groups and smooth structures of $(n-1)$-connected $2n$-manifolds

Let $M^{2n}$ denote a closed $(n-1)$-connected smoothable topological $2n$-manifold. We show that the group $\mathcal{C}(M^{2n})$ of concordance classes of smoothings of $M^{2n}$ is isomorphic to the group of smooth homotopy spheres $\overlineΘ_{2n}$ for $n=4$ or $5$, the concordance inertia group $I_c(M^{2n})=0$ for $n=3$, $4$, $5$ or $11$ and the homotopy inertia group $I_h(M^{2n})=0$ for $n=4$. On the way, following Wall's approach \cite{Wal67} we present a new proof of the main result in \cite{KS07}, namely, for $n=4$, $8$ and $H^{n}(M^{2n};\mathbb{Z})\cong \mathbb{Z}$, the inertia group $I(M^{2n})\cong \mathbb{Z}_2$. We also show that, up to orientation-preserving diffeomorphism, $M^{8}$ has at most two distinct smooth structures; $M^{10}$ has exactly six distinct smooth structures and then show that if $M^{14}$ is a $π$-manifold, $M^{14}$ has exactly two distinct smooth structures.

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arXiv2015

The classification of smooth structures on a homotopy complex projective space

We classify, up to diffeomorphism, all closed smooth manifolds homeomorphic to the complex projective $n$-space $\mathbb{C}\textbf{P}^n$, where $n=3$ and $4$. Let $M^{2n}$ be a closed smooth $2n$-manifold homotopy equivalent to $\mathbb{C}\textbf{P}^n$. We show that, up to diffeomorphism, $M^{6}$ has a unique differentiable structure and $M^{8}$ has at most two distinct differentiable structures. We also show that, up to concordance, there exist at least two distinct differentiable structures on a finite sheeted cover $N^{2n}$ of $\mathbb{C}\textbf{P}^n$ for $n=4, 7$ or $8$ and six distinct differentiable structures on $N^{10}$.

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arXiv2015

Smooth and PL-Rigidity Problems on Locally Symmetric Spaces

This is a survey on known results and open problems about Smooth and PL-Rigidity Problem for negatively curved locally symmetric spaces. We also review some developments about studying the basic topological properties of the space of negatively curved Riemannian metrics and the Teichmuller space of negatively curved metrics on a manifold.

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arXiv2015

Farrell-Jones spheres and inertia groups of complex projective spaces

We introduce and study a new class of homotopy spheres called Farrell-Jones spheres. Using Farrell-Jones sphere we construct examples of closed negatively curved manifolds $M^{2n}$, where $n=7$ or $8$, which are homeomorphic but not diffeomorphic to complex hyperbolic manifolds, thereby giving a partial answer to a question raised by C.S. Aravinda and F.T. Farrell. We show that every exotic sphere not bounding a spin manifold (Hitchin sphere) is a Farrell-Jones sphere. We also discuss the relationship between inertia groups of $\mathbb{C}\mathbb{P}^n$ and Farrell-Jones spheres.

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arXiv2015

Inertia groups of high dimensional complex projective spaces

For a complex projective space the inertia group, the homotopy inertia group and the concordance inertia group are isomorphic. In complex dimension 4n+1, these groups are related to computations in stable cohomotopy. Using stable homotopy theory, we make explicit computations to show that the inertia group is non-trivial in many cases. In complex dimension 9, we deduce some results on geometric structures on homotopy complex projective spaces and complex hyperbolic manifolds.

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