arXiv subjects

Hyeontae Jang

Publications and source records attributed to Hyeontae Jang.

7 recordsLinked to original sources

arXiv2026

On the toric lifting properties for simplicial $3$-spheres

We study the lifting problem for mod $2$ characteristic maps over simplicial $3$-spheres. Using a bad-block partition of the universal complex $X(\mathbb{Z}_2^4)$, we prove an avoidance criterion for liftability. We show that every simplicial $3$-sphere with at most $20$ vertices has the toric lifting property. We also obtain image-size and join-type results, and prove sharpness of the image-size bound in the universal-complex sense.

math.AT↗

arXiv2026

Small covers as pullbacks from the simplex

We introduce and study small covers that are pullbacks from the simplex, extending pullbacks from the linear model. Our main result gives several equivalent characterizations of this class, including torsion-freeness of odd-degree integral cohomology, vanishing of the first Steenrod square on even-degree mod $2$ cohomology, and relations among integral and mod $2$ Betti numbers.

math.AT↗

arXiv2025

Complete non-singular toric varieties with Picard number 4

We classify all complete non-singular toric varieties with Picard number four via a combinatorial framework based on fanlike simplicial spheres and characteristic maps. This classification yields $59$ fanlike seeds with Picard number four, along with all toric manifolds supported by them. As a consequence, we resolve a conjecture of Gretenkort, Kleinschmidt, and Sturmfels by presenting the first known examples of toric manifolds supported by neighborly polytopes. We also answer a question of Batyrev concerning minimal non-faces of such spheres.

math.AG↗

arXiv2024

Stellar subdivisions, wedges and Buchstaber numbers

A seed is a PL sphere that is not obtainable by a wedge operation from any other PL sphere. In this paper, we study two operations on PL spheres, known as the stellar subdivision and the wedge, that preserve the maximality of Buchstaber numbers and polytopality. We construct a new polytopal toric colorable seed from these two operations. As a corollary, we prove that the toric colorable seed inequality established by Choi and Park is tight.

math.CO↗

arXiv2024

Toric wedge induction and toric lifting property for piecewise linear spheres with a few vertices

Let $K$ be an $(n-1)$-dimensional piecewise linear sphere on $[m]$, where $m\leq n+4$. There are a canonical action of $m$-dimensional torus $T^m$ on the moment-angle complex $\mathcal{Z}_K$, and a canonical action of $\mathbb{Z}_2^m$ on the real moment-angle complex $\mathbb{R}\mathcal{Z}_K$, where $\mathbb{Z}_2$ is the additive group with two elements. We prove that any subgroup of $\mathbb{Z}_2^m$ acting freely on $\mathbb{R}\mathcal{Z}_K$ is induced by a subtorus of $T^m$ acting freely on $\mathcal{Z}_K$. The proof primarily utilizes a suitably modified method of toric wedge induction and the combinatorial structure of a specific binary matroid of rank $4$.

math.AT↗

arXiv2024

The characterization of $(n-1)$-spheres with $n+4$ vertices having maximal Buchstaber number

We present a computationally efficient algorithm that is suitable for graphic processing unit implementation. This algorithm enables the identification of all weak pseudo-manifolds that meet specific facet conditions, drawn from a given input set. We employ this approach to enumerate toric colorable seeds. Consequently, we achieve a comprehensive characterization of $(n-1)$-dimensional PL spheres with $n+4$ vertices that possess a maximal Buchstaber number. A primary focus of this research is the fundamental categorization of non-singular complete toric varieties of Picard number $4$. This classification serves as a valuable tool for addressing questions related to toric manifolds of Picard number $4$. Notably, we have determined which of these manifolds satisfy equality within an inequality regarding the number of minimal components in their rational curve space. This addresses a question posed by Chen, Fu, and Hwang in 2014 for this specific case.

math.GT↗

arXiv2022

Strong Cohomological rigidity of Bott manifolds

We show that the strong cohomological rigidity conjecture for Bott manifolds is true. Namely, any graded cohomology ring isomorphism between two Bott manifolds is induced by a diffeomorphism.

math.AT↗