arXiv subjects

Navnath Daundkar

Publications and source records attributed to Navnath Daundkar.

29 records · Page 2Linked to original sources

arXiv2025

Real Bott manifold structure of $n$-dimensional Klein bottle and its rational Betti numbers

Donald Davis initiated the study of an $n$-dimensional analogue of the Klein bottle. This generalized Klein bottle occurs as a moduli space of planar polygons for a certain choice of side lengths. In this paper, we show that the $n$-dimensional Klein bottle is a real Bott manifold and determine the corresponding Bott matrix. We determine the small cover structure on two other classes of moduli spaces of planar polygons. As an application, we compute the rational Betti numbers of these spaces using a formula, due to Suciu and Trevisan.

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arXiv2025

Sequential parametrized topological complexity of group epimorphisms

We introduce and study the sequential analogue of Grant's parametrized topological complexity of group epimorphisms, which generalizes the sequential topological complexity of groups. We derive bounds for sequential parametrized topological complexity based on the cohomological dimension of certain subgroups, thereby extending the corresponding bounds for sequential topological complexity of groups. We also obtain sequential analogs of (new) lower bounds on parametrized topological complexity of epimorphisms which are recently obtained by Espinosa Baro, Farber, Mescher and Oprea. Finally, we utilize these results to provide alternative computations for the sequential parametrized topological complexity of planar Fadell-Neuwirth fibrations.

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arXiv2025

Parametrized homotopic distance

We introduce the concept of parametrized homotopic distance, extending the classical notion of homotopic distance to the fibrewise setting. We establish its correspondence with the fibrewise sectional category of a specific fibrewise fibration and derive cohomological lower bounds and connectivity upper bounds under mild conditions. We also analyze the behavior of parametrized homotopic distance under compositions and products of fibrewise maps, along with its interaction with the triangle inequality. We establish several sufficient conditions for fibrewise $H$-spaces to admit a fibrewise division map and prove that their parametrized topological complexity equals their fibrewise unpointed Lusternik-Schnirelman category, extending Lupton and Scherer's theorem to the fibrewise setting. Additionally, we give sharp estimates for the parametrized topological complexity of a class fibrewise $H$-spaces which arises as sphere bundles with fibre $S^7$. Furthermore, we estimate the parametrized homotopic distance of fibre-preserving, fibrewise maps between fibrewise fibrations, in terms of the parametrized homotopic distance of the induced fibrewise maps between individual fibres, as well as the fibrewise unpointed Lusternik-Schnirelman category of the base space. Finally, we define and study a pointed version of parametrized homotopic distance, establishing cohomological bounds and identifying key conditions for its equivalence with the unpointed version, thus providing a finer classification of fibrewise homotopy invariants.

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arXiv2024

Equivariant parametrized topological complexity

In this paper, we define and study an equivariant analogue of Cohen, Farber and Weinberger's parametrized topological complexity. We show that several results in the non-equivariant case can be extended to the equivariant case. For example, we establish the fibrewise equivariant homotopy invariance of the sequential equivariant parametrized topological complexity. We obtain several bounds on sequential equivariant topological complexity involving equivariant category. We also obtain the cohomological lower bound and the dimension-connectivity upper bound on the sequential equivariant parametrized topological complexity. In the end we use these results to compute sequential equivariant parametrized topological complexity of equivariant Fadell-Neuwirth fibrations and some equivariant fibrations involving generalized projective product spaces.

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arXiv2024

Group actions and higher topological complexity of lens spaces

In this paper, we obtain an upper bound on the higher topological complexity of the total spaces of fibrations. As an application, we improve the usual dimensional upper bound on higher topological complexity of total spaces of some sphere bundles. We show that this upper bound on the higher topological complexity of the total spaces of fibrations can be improved using the notion of higher subspace topological complexity. We also show that the usual dimensional upper bound on the higher topological complexity of any path-connected space can be improved in the presence of positive dimensional compact Lie group action. We use these results to compute the exact value of higher topological complexity of higher dimensional lens spaces in many cases.

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arXiv2024

Building planar polygon spaces from the projective braid arrangement

The moduli space of planar polygons with generic side lengths is a smooth, closed manifold. It is known that these manifolds contain the real points of the moduli space of distinct points on the projective line as an open dense subset. Hence, such a polygon space is a compactification of this real moduli space. Kapranov showed that the real points of the Deligne-Mumford-Knudson compactification can be obtained from the projective Coxeter complex of type $A$ (equivalently, the projective braid arrangement) by iteratively blowing up along the minimal building set. In this paper we show that these planar polygon spaces can also be obtained from the projective Coxeter complex of type $A$ by performing an iterative cellular surgery along a sub-collection of the minimal building set. Interestingly, this sub-collection is determined by the combinatorial data associated with the length vector called the genetic code.

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arXiv2024

LS-category and topological complexity of Milnor manifolds and corresponding generalized projective product spaces

Milnor manifolds are a class of certain codimension-$1$ submanifolds of the product of projective spaces. In this paper, we study the LS-category and topological complexity of these manifolds. We determine the exact value of the LS-category and in many cases, the topological complexity of these manifolds. We also obtain tight bounds on the topological complexity of these manifolds. It is known that Milnor manifolds admit $\mathbb{Z}_2$ and circle actions. We compute bounds on the equivariant LS-category and equivariant topological complexity of these manifolds. Finally, we describe the mod-$2$ cohomology rings of some generalized projective product spaces corresponding to Milnor manifolds and use this information to compute the bound on LS-category and topological complexity of these spaces.

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arXiv2023

LS-category and topological complexity of several families of fibre bundles

In this paper, we study upper bounds for the topological complexity of the total spaces of some classes of fibre bundles. We calculate a tight upper bound for the topological complexity of an $n$-dimensional Klein bottle. We also compute the exact value of the topological complexity of $3$-dimensional Klein bottle. We describe the cohomology rings of several classes of generalized projective product spaces with $\mathbb{Z}_2$-coefficients. Then we study the LS-category and topological complexity of infinite families of generalized projective product spaces. We reckon the exact value of these invariants in many specific cases. We calculate the equivariant LS-category and equivariant topological complexity of several product spaces equipped with $\mathbb{Z}_2$-action.

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arXiv2023

Independence complexes of wedge of graphs

In this article, we introduce the notion of a wedge of graphs and provide detailed computations for the independence complex of a wedge of path and cycle graphs. In particular, we show that these complexes are either contractible or wedges of spheres.

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arXiv2022

The Borsuk-Ulam theorem for planar polygon spaces

The moduli space of planar polygons with generic side lengths is a closed, smooth manifold. Mapping a polygon to its reflected image across the $X$-axis defines a fixed-point-free involution on these moduli spaces, making them into free $\mathbb{Z}_2$-spaces. There are some important numerical parameters associated with free $\mathbb{Z}_2$-spaces, like index and coindex. In this paper, we compute these parameters for some moduli spaces of polygons. We also determine for which of these spaces a generalized version of the Borsuk-Ulam theorem hold. Moreover, we obtain a formula for the Stiefel-Whitney height in terms of the the genetic code, a combinatorial data associated with side lengths.

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arXiv2021

The moment polytope of the abelian polygon space

The moduli space of $n$ chains in the plane with generic side lengths that terminate on a fixed line is a smooth, closed manifold of dimension $n-1$. This manifold is also equipped with a locally standard action of $\mathbb{Z}_2^{n-1}$. The orbit space of this action is a simple polytope called the moment polytope. Interestingly, this manifold is also the fixed point set of an involution on a toric manifold known as the abelian polygon space. In this article we show that the moment polytope of the moduli space of chains is completely characterized by the combinatorial data, called the \emph{short code} of the length vector. We also classify aspherical chain spaces using a result of Davis, Januszkiewicz and Scott.

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