arXiv subjects

Anshu Agarwal

Publications and source records attributed to Anshu Agarwal.

6 recordsLinked to original sources

arXiv2026

A classification of semi-equivelar gems on the double torus

A \emph{semi-equivelar gem} of a PL $d$-manifold is a regular colored graph that represents the manifold and admits a regular embedding on a surface, such that the cyclic sequence of face degrees around each vertex is identical. In \cite{ab25, bb24}, semi-equivelar gems of PL $d$-manifolds embedded on surfaces with Euler characteristic $χ\geq -1$ were classified for $d\geq 2$. In this paper, we extend this classification to semi-equivelar gems embedded on the double torus. We show that any such gem must belong to one of the following thirty two types: $(4^6)$, $(4^5)$, $(6^4)$, $(4^3,6)$, $(4^3,8)$, $(4^3,12)$, $(4^2,6^2)$, $(4,6,4,6)$, $(4^2,8^2)$, $(4,8,4,8)$, $(8^3)$, $(10^3)$, $(6^2,8)$, $(6^2,10)$, $(6^2,12)$, $(6^2,18)$, $(10^2,4)$, $(12^2,4)$, $(16^2,4)$, $(8^2,6)$, $(12^2,6)$, $(4,6,14)$, $(4,6,16)$, $(4,6,18)$, $(4,6,20)$, $(4,6,24)$, $(4,6,36)$, $(4,8,10)$, $(4,8,12)$, $(4,8,16)$, $(4,8,24)$, and $(4,10,20)$. Furthermore, we provide explicit constructions of semi-equivelar gems realizing each of these types.

math.GT↗

arXiv2026

The Homotopy Types of the Independence and Perfect Matching Complexes of Möbius and Circular Ladder Graphs

The independence complex and perfect matching complex of a graph are simplicial complexes encoding, respectively, its independent sets and perfect matchings. Determining their homotopy types is generally difficult, with explicit descriptions known mainly for highly structured graph families. In this article, we determine the homotopy types of these complexes for the Möbius ladder graphs $M_{2n}$ and circular ladder graphs $\mathcal{C}_{2n}$. The Möbius ladder graphs $M_{2n}$ are highly symmetric cubic graphs obtained from a $2n$-cycle by joining opposite vertices, while the circular ladder graphs $\mathcal{C}_{2n}$ are the Cartesian products of an $n$-cycle and a path of length one. We show that $\operatorname{Ind}(M_{2n})$ and $\operatorname{Ind}(\mathcal{C}_{2n})$ have the homotopy type of wedges of spheres, with the numbers and dimensions of the spheres exhibiting periodic behavior according to $n$ modulo $4$. For the perfect matching complex $\mathcal{M}_p(M_{2n})$, its homotopy type is a wedge of two copies of $\mathbb{S}^{(n-2)/2}$ when $n$ is even, while for odd $n$ it has the homotopy type of a wedge of spheres whose numbers and dimensions depend periodically on $n$ modulo $6$. The perfect matching complex $\mathcal{M}_p(\mathcal{C}_{2n})$ is contractible for odd $n$, whereas for even $n$ its homotopy type is a wedge of spheres, with the numbers and dimensions determined periodically by $n$ modulo $6$. Thus, we obtain explicit homotopy types for the independence and perfect matching complexes of two highly symmetric families of cubic graphs, which are also relevant in crystallization theory and the combinatorial representation of PL manifolds.

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arXiv2026

Minimal simplicial degree $d$ self-maps of $\mathbb{S}^{n-1}\times \mathbb{S}^1$

The degree of a map between orientable manifolds is a fundamental concept in topology, providing important information about the structure of manifolds and the behavior of maps between them. A simplicial cell complex $K$ is called a \emph{colored triangulation} of a closed PL $n$-manifold $M$ if the $1$-skeleton of $K$ admits a proper vertex-coloring with $n+1$ colors and $|K|$ is PL-homeomorphic to $M$. In this article, we construct, for every $d \in \mathbb{Z}$ and $n \geq 2$, a degree $d$ simplicial map from a $(2(n+1)\max\{|d|,1\})$-facet colored triangulation of $\mathbb{S}^{n-1} \times \mathbb{S}^1$ to the standard $2(n+1)$-facet colored triangulation of $\mathbb{S}^{n-1} \times \mathbb{S}^1$. Additionally, for every $d \in \mathbb{Z}$ and $n \geq 2$, we construct a degree $d$ simplicial map from a $(2\max\{|d|,1\})$-facet colored triangulation of $\mathbb{S}^n$ to the standard $2$-facet colored triangulation of $\mathbb{S}^n$. For $M = \mathbb{S}^{n-1} \times \mathbb{S}^1$ and $\mathbb{S}^n$, with $n \geq 2$, these simplicial degree $d$ self-maps of $M$ are minimal with respect to their standard colored triangulations, in the sense that there does not exist a colored triangulation $K$ of $M$ with fewer facets than the constructed one that admits a simplicial map $f : K \to K'$ of degree $d$, where $K'$ denotes the standard colored triangulation of $M$.

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arXiv2026

Crystallizations of small covers over the $n$-simplex $Δ^n$ and the prism $Δ^{n-1} \times I$

A crystallization of a PL manifold is an edge-colored graph that corresponds to a contracted triangulation of the manifold, facilitating the study of its topological and combinatorial properties. A small cover over a simple convex $n$-polytope $P^n$ is a closed $n$-manifold with a locally standard $\mathbb{Z}_2^n$-action such that its orbit space is homeomorphic to $P^n$. In this article, we study the crystallizations of small covers over the $n$-simplex $Δ^n$ and the prism $Δ^{n-1} \times I$. It is known that the small cover over the $n$-simplex $Δ^n$ is $\mathbb{RP}^n$. For every $n\geq 2$, we prove that $\mathbb{RP}^n$ has a unique $2^n$-vertex crystallization. We also demonstrate that there are exactly $1 + 2^{n-1}$ D-J equivalence classes of small covers over the prism $Δ^{n-1} \times I$, where $n\geq 3$. For each $\mathbb{Z}_2$-characteristic function of $Δ^{n-1} \times I$, we construct a $2^{n-1}(n+1)$-vertex crystallization of the small cover $M^n(λ)$ with regular genus $1 + 2^{n-4}(n^2 - 2n - 3)$, where $n\geq 4$. The regular genus of closed PL $n$-manifolds extends the notions of the genus of surfaces and the Heegaard genus of 3-manifolds to higher dimensions. In this article, we construct four orientable and four non-orientable $\mathbb{RP}^3$-bundles over $\mathbb{S}^1$ up to D-J equivalence, each with regular genus $6$. Although the four orientable (resp. non-orientable) small covers are not D-J equivalent, we show that they are PL homeomorphic.

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arXiv2025

Regular genus of $\mathbb{S}^2 \times \mathbb{S}^1 \times \mathbb{S}^1$, $4$-torus, and small covers over $Δ^2 \times Δ^2$

A crystallization of a PL manifold is an edge-colored graph encoding a contracted triangulation of the manifold. The concept of regular genus generalizes the notions of surface genus and Heegaard genus for 3-manifolds to higher-dimensional closed PL manifolds. The regular genus of a PL manifold is a PL invariant. Determining the regular genus of a closed PL $n$-manifold remains a fundamental challenge in combinatorial topology. In this article, we first resolve a conjecture by proving that the regular genus of $\mathbb{S}^2 \times \mathbb{S}^1 \times \mathbb{S}^1$ is 6. Additionally, we determine that the regular genus of $\mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{S}^1$ is 16. We also present some observations related to the regular genus of the $n$-dimensional torus and conjecture that the regular genus of $\mathbb{S}^1 \times \mathbb{S}^1 \times \cdots \times \mathbb{S}^1$ ($n$ times) is $1+\frac{(n+1)! \ (n-3)}{8}$, for $n\ge 5$. Then, we investigate the regular genus of small covers. Small covers are closed $n$-manifolds admitting a locally standard $\mathbb{Z}_2^n$-action with orbit space homeomorphic to a simple convex polytope $P^n$. For the polytope $P = Δ^2 \times Δ^2$, we classify all the small covers up to Davis-Januszkiewicz (D-J) equivalence and show that there are exactly seven such covers. Among these, one is $\mathbb{RP}^2 \times \mathbb{RP}^2$, while the others are $\mathbb{RP}^2$-bundles over $\mathbb{RP}^2$. Remarkably, each of these seven small covers has the regular genus 8. Results in this article provide explicit regular genus values for several important 4-manifolds, offering new insights and tools for future work in combinatorial topology.

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arXiv2024

A classification of semi-equivelar gems of PL $d$-manifolds on the surface with Euler characteristic $-1$

A semi-equivelar gem of a PL $d$-manifold is a regular colored graph that represents the PL $d$-manifold and regularly embeds on a surface, with the property that the cyclic sequence of degrees of faces in the embedding around each vertex is identical. In \cite{bb24}, the authors classified semi-equivelar gems of PL $d$-manifolds embedded on surfaces with Euler characteristics greater than or equal to zero. In this article, we focus on classifying semi-equivelar gems of PL $d$-manifolds embedded on the surface with Euler characteristic $-1$. We prove that if a semi-equivelar gem embeds regularly on the surface with Euler characteristic $-1$, then it belongs to one of the following types: $(8^3), (6^2,8), (6^2,12), (10^2,4), (12^2,4),$ $ (4,6,14), (4,6,16), (4,6,18), (4,6,24), (4,8,10), (4,8,12),$ or $(4,8,16)$. Furthermore, we provide constructions that demonstrate the existence of such gems for each of the aforementioned types.

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