arXiv subjects

Ankur Sarkar

Publications and source records attributed to Ankur Sarkar.

9 recordsLinked to original sources

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

A Precise Treatment of Soft Quotient Topology and Soft Covering Maps

We develop a foundational theory of soft quotient topology, providing a systematic approach to quotient constructions in soft topological spaces. We establish the universal property of soft quotient topology and investigate the relationship between global soft topology and its parametric slices, noting that slice-wise quotient behaviour is not sufficient to characterise soft quotients. The usefulness of the framework is illustrated through the construction of soft quotient spaces, together with a study of soft group actions and their orbit spaces. We propose a precise definition of soft covering maps that resolves inconsistencies found in the existing literature. Finally, to illustrate the applicability of our framework, we discuss a potential application in multi-agent motion planning, showing how it can significantly reduce combinatorial complexity under parametric uncertainty.

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arXiv2026

Relative Smooth Surgery Structure Sets of Thickenings of the Cayley Projective Plane and Applications

We compute the relative smooth surgery structure sets of the thickenings $\mathbb{OP}^{2}\times\mathbb{D}^{k}$ of the Cayley projective plane $\mathbb{OP}^{2}$ for every $k\geq 1$ with $k\equiv 0\pmod 4$, by determining the corresponding normal invariants and surgery obstruction map. We show that the latter is not surjective and determine the $2$-adic valuation of the generator of its image. As applications, we construct infinitely many pairwise non-homeomorphic closed smooth manifolds of dimension $16+k$, homotopy equivalent to $\mathbb{OP}^{2}\times\mathbb{S}^{k}$ and distinguished by their Pontryagin numbers; we compute the rational homotopy groups of the block diffeomorphism group $\widetilde{\operatorname{Diff}}(\mathbb{OP}^{2})$ in every degree congruent to $3$ modulo $4$; and we construct smooth $\mathbb{OP}^{2}$-bundles over $\mathbb{S}^{4}$, $\mathbb{S}^{8}$, and $\mathbb{S}^{12}$ whose total spaces have non-vanishing $\widehat{\mathfrak{A}}$-genus. These bundles yield elements of infinite order in the homotopy groups of the spaces of metrics of positive sectional, Ricci, and scalar curvature on $\mathbb{OP}^{2}$ in degrees $3$, $7$, and $11$.

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arXiv2026

Countable Fan Tightness and Selection Games in Group-Valued Function Spaces

Game-theoretic characterizations of selection principles provide a powerful framework for analyzing covering properties through strategic interactions. For a Tychonoff space $X$ and a non-trivial metrizable arc-connected topological group $G$, we prove that Player~II has a winning strategy in the $\Omega$-Menger game on $X$ if and only if Player~II has a winning strategy in the countable fan tightness game on $C_p(X, G)$ at the identity function. The analogous equivalence is established between the $\Omega$-Rothberger game on $X$ and the countable strong fan tightness game on $C_p(X, G)$ at the identity function. These results extend the game-theoretic characterizations of Clontz from $G = \mathbb{R}$ to arbitrary metrizable arc-connected groups, and lift the selection-principle equivalences of Ko\v{c}inac to the game-theoretic setting. As consequences, we establish that the game-theoretic tightness properties of $C_p(X,G)$ are independent of $G$, preserved under $G$-equivalence, and remain valid for Markov strategies.

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arXiv2026

Prime Density and Classification of Mac\'ias Spaces over Principal Ideal Domains

Recently, the Mac\'ias topology has been generalized over integral domains that are not fields, to furnish a topological proof of the infinitude of prime elements under the assumption that the set of units is finite or not open. In this article, we remove this cardinality assumption completely by using the Jacobson radical. We prove that in any semiprimitive integral domain, the group of units is not open in the Mac\'ias topology. Consequently, for a principal ideal domain, this gives an equivalence between the triviality of the Jacobson radical, the density of the set of prime elements, and the group of units not being open in the Mac\'ias topology. Furthermore, we completely characterize when Mac\'ias spaces over different infinite principal ideal domains are homeomorphic in terms of cardinalities of certain subsets of the domains. As an application we resolve an open problem concerning homeomorphism of Mac\'ias spaces over countably infinite semiprimitive principal ideal domains.

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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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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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arXiv2025

Smooth Structures on the product of 3-connected 8-manifolds with spheres

Let $M$ be a closed, 3-connected, 8-dimensional smooth manifold. In this paper, we compute the concordance inertia group of the product manifold $M\times\mathbb{S}^k$ for $1\leq k\leq 14$ and classify all smooth manifolds homeomorphic to $M\times\mathbb{S}^k,$ up to concordance for $1\leq k\leq 10.$ Moreover, we provide a diffeomorphism classification of smooth manifolds homeomorphic to $M\times\mathbb{S}^1,$ where $H^4(M;\mathbb{Z})=\mathbb{Z}.$

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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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