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Ramandeep Singh Arora

Publications and source records attributed to Ramandeep Singh Arora.

6 recordsLinked to original sources
arXiv2026

Equivariant principal bundles over toric Deligne--Mumford stacks

Let $\mathscr{X}(\boldsymbolΣ)$ be the toric Deligne--Mumford stack with stacky torus $\mathscr{T}$ associated to the stacky fan $\boldsymbolΣ = (N,Σ, β)$. A toric principal $H$-bundle over $\mathscr{X}(\boldsymbolΣ)$ is a principal $H$-bundle $\mathscr{P}$ equipped with a $\mathscr{T}$-action lifting the $\mathscr{T}$-action on $\mathscr{X}(\boldsymbolΣ)$, such that the $\mathscr{T}$-action and the $H$-action on $\mathscr{P}$ commute. For a connected reductive algebraic group $H$ over $\mathbb{C}$, we classify the isomorphism classes of framed toric principal $H$-bundles over $\mathscr{X}(\boldsymbolΣ)$ in terms of piecewise linear maps from $|Σ|$ to the cone over the Tits building of $H$. Using this classification, we describe the equivariant automorphism group of a toric principal $H$-bundle, and give a necessary and sufficient condition for an equivariant reduction of the structure group. We also show that every toric principal $\mathrm{GL}(r)$-bundle over the weighted projective stack $\mathbb{P}(w_0,\dots,w_n)$ splits equivariantly whenever $r < n$, and that every toric principal $H$-bundle over the weighted stacky projective line $\mathbb{P}(a,b)$ splits equivariantly when $H$ is a connected reductive algebraic group.

math.AG↗
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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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-Guzmán--González (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

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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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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