Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2606.15241

Filtered order complexes and magnitude homology of finite graded posets

Yoh Kitajima
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we study the family of subcomplexes of the order complexes of finite graded posets, defined via its rank function. We address three main topics. (1) We describe the general topological properties of these subcomplexes in relation to magnitude homology of graded posets. (2) For posets whose order complexes are simplicial subdivisions of closed manifolds, we show that the homology groups of these subcomplexes agree with that of the undelying manifold except for the top dimension, where it is a nontrivial free abelian group. (3) For shellable graded posets, we prove that each of the subcomplexes are also shellable. Moreover, in the case of geometric semilattices, we show that each subcomplexes are homotopy equivalent to a nontrivial wedge sums of spheres of the same dimension.

Explore related subjects

math.COmath.ATmath.MG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yoh Kitajima. 2026-06-24. Filtered order complexes and magnitude homology of finite graded posets. https://arxiv.org/abs/2606.15241

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Adjunctions, Box Products, and Forcing Families

Sidorenko's conjecture states that the number of copies of any given bipartite graph in another graph of given density is asymptotically minimized by a random graph. For bipartite graphs containing a cycle, the forcing conjecture further asserts that asymptotic equality characterizes quasi-random graphs. We establish an adjoint identity for a general class of graph-substitution operators and use it to obtain Sidorenko and forcing results for balanced blow-ups, subdivisions, Cartesian products, and strong products.

math.CO↗
arXiv2026

On the Cost Number of Graphs with Determining Number Two

A distinguishing vertex coloring of a graph $G$ is a vertex coloring such that only the identity automorphism of $G$ preserves the coloring. A graph is $2$-distinguishable if it admits a distinguishing vertex coloring with two colors, and its cost $ρ(G)$ is the minimum size of a color class in such a coloring. The determining number of a graph $G$, denoted by $Det(G)$, is the minimum size of a subset $S\subseteq V(G)$ such that only the trivial automorphism fixes every element of $S$ pointwise. Boutin (J. Combin. Math. Combin. Comput. 85: 161-171, 2013) asked if $ρ(G)$ and $Det(G)$ can be arbitrarily far apart. While the case for $Det(G) = 1$ is trivial, the answer remained unknown for $Det(G) \ge 2$. In this manuscript, we show that if $Det(G)=2$ then not only is $ρ(G)$ bounded, but in fact $ρ(G) \leq 4$. This is the first resolution of Boutin's question for any nontrivial fixed determining number. Moreover, for every fixed $Det(G)= n$, we construct examples giving a lower bound on any possible upper bound for $ρ(G)$ in terms of $n$.

math.CO↗