arXiv subjects

Navnath Daundkar

Publications and source records attributed to Navnath Daundkar.

At least 19 recordsLinked to original sources

arXiv2026

On generalized complex projective product spaces

Free circle action on manifolds has been explored in several articles. Gonzalez and Velasco considered some free circle actions on the finite product of spheres. In this paper, we introduce generalized complex projective product spaces, extending their definition and the concept of Dold manifolds. This produces infinitely many different classes of new smooth manifolds. First, we study the integral cohomology rings and stable tangent bundles on certain generalized complex-projective product spaces. Then, we discuss the product inequality for strongly equivaraint TC. We exhibit the closeness of the lower and upper bounds for the LS-category and topological complexity of several classes of generalized complex projective product spaces. In many cases, we compute the exact value of the LS-category.

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arXiv2026

Equivariant Relative Sectional Category and Induced Invariants

The relative sectional category, introduced by Gonz\'alez, Grant, and Vandembroucq for fibrations and later extended by Garc\'ia-Calcines to arbitrary maps, provides a common framework encompassing several numerical homotopy invariants, including the Lusternik--Schnirelmann category, the topological complexity of a map, and homotopic distance. In this paper, we introduce and study the equivariant analogue of the relative sectional category for $G$-maps. We establish its fundamental homotopy-theoretic properties, including comparison, product, and composition inequalities, as well as its behavior under changes of domain and codomain. As applications, we introduce and investigate equivariant analogues of the topological complexity of a map, in the sense of Scott and Murillo--Wu, and the equivariant Lusternik--Schnirelmann category of a map. Several examples are provided to illustrate the theory and demonstrate that these invariants extend the corresponding classical equivariant notions.

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arXiv2026

On the Sequential topological complexity of directed (parametrized) motion planning algorithms

We introduce sequential analogues of directed (parametrized) topological complexity, in the context of motion planning problems requiring a system to traverse a prescribed sequence of intermediate states while respecting directed dynamics and varying external parameters. We develop their basic theory, establish fundamental properties, and compute them for several classes of examples. Our computations show, in particular, that distinct directed structures on the same underlying space can have different values of this invariant.

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arXiv2026

Monoidal and symmetrized parametrized topological complexity

We introduce and study monoidal and symmetrized versions of parametrized topological complexity. First, we develop parametrized analogues of the monoidal topological complexity theories of Iwase--Sakai, Aguilar-Guzm\'an--Gonz\'alez (based on the Fadell--Husseini approach), and Dranishnikov. We investigate the parametrized Iwase--Sakai conjecture and provide sufficient conditions under which it holds. We then introduce symmetrized parametrized topological complexity and combine it with the monoidal perspective to define monoidal symmetrized parametrized topological complexity, showing that the resulting notions agree. Finally, we compute these invariants for Fadell--Neuwirth fibrations.

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arXiv2026

New Bounds on Distributional Sectional Category and Applications to Distributional Homotopic Distance

In this paper, we establish several new bounds for the distributional sectional category ($\mathrm{dsecat}$). We first prove Jauhari's conjecture, thereby establishing a cohomological lower bound for $\mathrm{dsecat}$ with arbitrary coefficients. We then obtain an analogous lower bound with rational coefficients by constructing a natural splitting of the map induced on cohomology by the diagonal inclusion into symmetric powers. As applications, we provide several computations of the distributional sectional category. We also establish multiplicative product and composition inequalities for $\mathrm{dsecat}$. Finally, we apply these results to the distributional homotopic distance recently introduced by Jauhari and Oprea, giving an equivalent formulation in terms of distributed homotopies and deriving new fibration and multiplicative triangle inequalities.

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arXiv2026

Discrete version of topological complexity of maps

We introduce and study discrete analogs of Scott's and Murillo-Wu's topological complexity of maps. We prove that these discrete analogs are contiguity invariants and are, in fact, equivalent. Furthermore, we establish the fundamental theoretical properties and computational aspects of the discrete topological complexity of simplicial maps.

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arXiv2026

On sequential versions of various parametrized invariants

In this paper, we introduce and study sequential versions of several fibrewise homotopy invariants, including parametrized topological complexity, parametrized (subspace) homotopic distance. We investigate their basic properties, establish relationships among them, and compare them with the corresponding classical homotopical invariants.

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arXiv2026

On the $m$-dimensional sectional category and induced invariants

In this paper, we systematically study the $m$-dimensional sectional category of a fibration, introduced by Schwarz as an approximating invariant for the sectional category. We develop the basic theory of this invariant, establish its fundamental properties, and show how it gives rise to a hierarchy of induced invariants, including the $m$-dimensional Lusternik--Schnirelmann category, the $m$-topological complexity, and the $m$-homotopic distance between maps. We further investigate the relationships between these $m$-dimensional invariants and their classical analogues, present a variety of examples in which these invariants are computed, and illustrate when they agree with or differ from their classical counterparts. We also introduce the notion of $m$-cohomological distance and study its interaction with the $m$-homotopic distance. Finally, we prove a Bochner-type theorem for $\mathrm{secat}_1$, extending the corresponding theorem of Oprea and Strom for $\mathrm{cat}_1$. We also establish a $\mathrm{cat}_m$ version of Oprea's improvement of Bochner's theorem.

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arXiv2025

Equivariant homotopic distance

We introduce and study the notion of \emph{equivariant homotopic distance} $D_G(f,g)$ between $G$-maps $f,g \colon X \to Y$. We show that the equivariant Lusternik-Schnirelmann category and the equivariant topological complexity are particular cases of this notion. This invariant also connects naturally with the equivariant sectional category. What makes $D_G$ distinctive, however, is that it provides a flexible framework centered on pairs of maps, within which one can derive results that are not immediate from the general setting. In particular, we establish its basic properties, including homotopy invariance and a categorical proof of the triangle inequality valid in the equivariant context. We also obtain cohomological and dimension-connectivity bounds, and analyze structural applications to Hopf $G$-spaces and equivariant fibrations.

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arXiv2025

Higher topological complexity of planar polygon spaces having small genetic codes

We study the higher (sequential) topological complexity, a numerical homotopy invariant for the planar polygon spaces. For these spaces with a small genetic codes and dimension $m$, Davis showed that their topological complexity is either $2m$ or $2m+1$. We extend these bounds to the setting of higher topological complexity. In particular, when $m$ is power of $2$, we show that the $k$-th higher topological complexity of these spaces is either $km$ or $km+1.$

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arXiv2025

On the complexity of parametrized motion planning algorithms

We study a probabilistic variant of the r-th sequential parametrized topological complexity, which bounds this classical invariant from below and measures the difficulty in constructing permissive parametrized motion planning algorithms. On one hand, we use cohomology to show that this new invariant behaves similarly to the classical invariant on Fadell-Neuwirth fibrations and oriented sphere bundles; on the other hand, we use equivariant homotopy theory to prove that its behavior is wildly different on bundles whose fibers are real projective spaces and whose structure groups are special orthogonal groups. We also explore several other features of our invariant and its relationships with various other invariants motivated by topological robotics.

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arXiv2025

Higher (equivariant) topological complexity of Milnor manifolds

J. Milnor introduced a specific class of codimension-$1$ submanifolds in the product of projective spaces, known as Milnor manifolds. This paper establishes precise bounds on the higher topological complexity of these manifolds and provides exact values for this invariant for numerous Milnor manifolds. Furthermore, we improve the upper bounds on the higher equivariant topological complexity. As an application, we obtain sharper bounds on the higher equivariant topological complexity of Milnor manifolds with free $\mathbb{Z}_2$ and $S^1$-actions.

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arXiv2025

Sectional category with respect to group actions and sequential topological complexity of fibre bundles

Let $X$ be a $G$-space. In this paper, we introduce the notion of sectional category with respect to $G$. As a result, we obtain $G$-homotopy invariants: the LS category with respect to $G$, the sequential topological complexity with respect to $G$ (which is same as the weak sequential equivariant topological complexity $\mathrm{TC}_{k,G}^w(X)$ in the sense of Farber and Oprea), and the strong sequential topological complexity with respect to $G$, denoted by $\mathrm{cat}_G^{\#}(X)$, $\mathrm{TC}_{k,G}^{\#}(X)$, and $\mathrm{TC}_{k,G}^{\#,*}(X)$, respectively. We explore several relationships among these invariants and well-known ones, such as the LS category, the sequential (equivariant) topological complexity, and the sequential strong equivariant topological complexity. In one of our main results, we give an additive upper bound for $\mathrm{TC}_k(E)$ for a fibre bundle $F \hookrightarrow E \to B$ with structure group $G$ in terms of certain motion planning covers of the base $B$ and the invariant $\mathrm{TC}_{k,G}^{\#,*}(F)$ or $\mathrm{cat}_{G^k}^{\#}(F^k)$, where the fibre $F$ is viewed as a $G$-space. As applications of these results, we give bounds on the sequential topological complexity of generalized projective product spaces and mapping tori.

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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

On the topological complexity of directed parametrized motion planning

We introduce and study a parametrized analogue of the directed topological complexity, originally developed by Goubault, Farber, and Sagnier. We establish the fibrewise basic dihomotopy invariance of directed parametrized topological complexity and explore its relationship with the parametrized topological complexity. In addition, we introduce the concept of the directed Lusternik-Schnirelmann (LS) category, prove its basic dihomotopy invariance, and investigate its connections with both directed topological complexity and directed parametrized topological complexity. We further investigate additional properties of our invariant and examine its connections with several other invariants that arise naturally in the context of topological robotics. Moreover, we compute the directed parametrized topological complexity of the Hopf fibrations and the Fadell-Neuwirth fibrations having specific directed fibration structures.

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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 and invariant parametrized topological complexity

For a $G$-equivariant fibration $p \colon E\to B$, we introduce and study the invariant analogue of Cohen, Farber and Weinberger's parametrized topological complexity, called the invariant parametrized topological complexity. This notion generalizes the invariant topological complexity introduced by Lubawski and Marzantowicz. When $G$ is a compact Lie group acting freely on $E$, we show that the invariant parametrized topological complexity of the $G$-fibration $p \colon E\to B$ coincides with the parametrized topological complexity of the induced fibration $\overline{p} \colon \overline{E} \to \overline{B}$ between the orbit spaces. Furthermore, we compute the invariant parametrized topological complexity of equivariant Fadell-Neuwirth fibrations, which measures the complexity of motion planning in the presence of obstacles with unknown positions, where the order of their placement is irrelevant. In addition, we study the equivariant sectional category and the equivariant parametrized topological complexity, which serve as essential tools for obtaining several results in this paper.

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