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Younghan Yoon

Publications and source records attributed to Younghan Yoon.

8 recordsLinked to original sources
arXiv2026

On small covers over Bier spheres

The Bier sphere of a simplicial complex $K$ is defined as the deleted join of $K$ and its combinatorial Alexander dual. We focus on the class of Bier spheres of the skeleta of a simplex. Since these Bier spheres are known to be polytopal, they give rise to small covers. We classify small covers over these Bier spheres up to Davis--Januszkiewicz equivalence. As applications, for all $m \geq 4$, we determine the homeomorphism types of small covers over the Bier spheres of the $0$-skeleton and the $(m-3)$-skeleton of an $(m-1)$-simplex. For the remaining cases $0<r<m-3$, we compute their rational Betti numbers.

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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.

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arXiv2026

A decomposition of graph a-numbers

We study the $a$-sequence $(a_0(G), a_1(G), \cdots)$ of a finite simple graph $G$, defined recursively through a combinatorial rule and known to coincide with the sequence of rational Betti numbers of the real toric variety associated with $G$. In this paper, we establish a combinatorial and topological decomposition formula for the $a$-sequence. As an application, we show that the $a$-sequence is monotone under graph inclusion; that is, $a_i(G) \geq a_i(H)$ for all $i \geq 0$ whenever $H$ is a subgraph of $G$, and obtain the lower and upper bounds of $a_i$-numbers. We also prove that the $a$-sequence is unimodal in $i$ for a broad class of graphs $G$, including those with a Hamiltonian circuit or a universal vertex. These results provide a new class of topological spaces whose Betti number sequences are unimodal but not necessarily log concave, contributing to the study of real loci in algebraic geometry.

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arXiv2025

Full subcomplexes of Bier spheres

Full subcomplexes of a simplicial complex encode essential structure for understanding the complex itself. For a simplicial complex $K$, possibly with a ghost vertex, the Bier sphere of $K$ is a simplicial sphere obtained as the deleted join of $K$ and its combinatorial Alexander dual. In this paper, we determine the homotopy types of all full subcomplexes of Bier spheres. As applications, we provide a formula for the bigraded Betti numbers of the Bier sphere of $K$ in terms of full subcomplexes of $K$, and we explicitly describe the cohomology of real toric manifolds associated with Bier spheres.

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arXiv2025

Real toric manifolds associated with chordal nestohedra

This paper investigates the rational Betti numbers of real toric manifolds associated with chordal nestohedra. We consider the poset topology of a specific poset induced from a chordal building set, and show its EL-shellability. Based on this, we present an explicit description using alternating $\mathcal{B}$-permutations for a chordal building set $\mathcal{B}$, transforming the computing Betti numbers into a counting problem. This approach allows us to compute the $a$-number of a finite simple graph through permutation counting when the graph is chordal. In addition, we provide detailed computations for specific cases such as real Hochschild varieties corresponding to Hochschild polytopes.

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arXiv2025

Cohomology of type $B$ real permutohedral varieties

Type $A$ and type $B$ permutohedral varieties are classic examples of mathematics, and their topological invariants are well known. This naturally leads to the investigation of the topology of their real loci, known as type $A$ and type $B$ real permutohedral varieties. The rational cohomology rings of type $A$ real permutohedral varieties are fully described in terms of alternating permutations. Until now, only rational Betti numbers of type $B$ real permutohedral varieties have been described in terms of $B$-snakes. In this paper, we explicitly describe the multiplicative structure of the cohomology rings of type $B$ real permutohedral varieties in terms of $B$-snakes.

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arXiv2024

The cohomology rings of real permutohedral varieties

A permutohedral variety is a remarkable object in various areas of mathematics, and its topological invariants are widely recognized. However, only little is known about a real permutohedral variety, that is, the real locus of a permutohedral variety. The rational Betti numbers of real permutohedral varieties were computed in terms of alternating permutations in 2012. In this paper, we provide explicit descriptions of the cohomology ring of real permutohedral varieties. In particular, we describe the multiplicative structure in terms of alternating permutations.

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