Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.29690

Discrete homotopy of graphs and $k$-parabolic subspace arrangements

So Yamagata
View original source ↗Read source document ↗Export metadata ↓

Abstract

Discrete homotopy theory is a combinatorial homotopy theory defined for discrete objects. Barcelo--Severs--White conjectured that the homotopy groups of complements of $k$-parabolic subspace arrangements are isomorphic to the discrete homotopy groups of graphs associated with the Coxeter complexes. We show that this correspondence fails in general by giving a counterexample in degree two for the $4$-parabolic subspace arrangement of type $A_5$.

Explore related subjects

math.ATmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

So Yamagata. 2026-10-21. Discrete homotopy of graphs and $k$-parabolic subspace arrangements. https://arxiv.org/abs/2610.29690

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

KEEP EXPLORING

Related papers

arXiv2026

Low-degree mod 2 cohomology of classifying spaces of $G_2$-gauge groups

Let $\mathcal G_k$ be the gauge group of the principal $G_2$-bundle over $S^4$ classified by $k\inπ_4(BG_2)\cong\mathbb Z$. In this paper, we determine the first positive fibre row in the mod $2$ Serre spectral sequence of $Ω^3_0G_2\to B\mathcal G_k\xrightarrow{p_k}BG_2$. The fibre cohomology vanishes in degrees $1$ through $4$ and has a unique nonzero class $u_5$ in degree $5$. We prove that its transgression is $d_6(u_5)=(k\bmod 2)x_6$. The proof uses the Stiefel--Whitney classes of the seven-dimensional representation and maps of evaluation fibrations induced by the pinch map of $S^4$. The same comparison maps show that the possible incoming differential $d_4:H^8(Ω^3_0G_2;\mathbb F_2)\to\mathbb F_2\{x_4u_5\}$ vanishes. Consequently, the classes $u_5$ and $x_4u_5$ survive precisely when $k$ is even. In degree six, a nonzero linear functional $λ:H^6(Ω^3_0G_2;\mathbb F_2)\to\mathbb F_2$ determines the next transgression by $d_7(z)=(k\bmod2)λ(z)x_7$. Both $d_7$ maps to the degree-seven and degree-eleven base classes vanish for even $k$ and are surjective for odd $k$; in particular, $p_k^*x_7\ne0$ exactly when $k$ is even. We derive further forced vanishings, cohomological consequences, and Steenrod-square relations, and delimit the remaining differentials with source of total degree at most $10$. The calculation is independent of the exact order of the fundamental Samelson product.

math.AT↗
arXiv2026

Quasimonophobic graphs and degree spectral sequences in discrete cubical homology

We introduce the degree filtration on the discrete cubical chain complex of a graph, defined in terms of the maximal injective dimension of the facets of singular $n$-cubes, and study the degree spectral sequence which arises from this filtration. This spectral sequence interpolates between the discrete cubical homology of a graph $H_n(G)$ and the injective homology $H_n^{inj}(G)$, a variant of the discrete cubical homology based on injective singular cubes. Building on the work of Greene and the first author, we introduce the combinatorial condition of quasimonophobicity on graphs, and show quasimonophobicity implies both the vanishing of the degree spectral sequence in certain bidegrees, and implies $H_n^{inj}(G)$ is isomorphic to the homology of the CW complex obtained by ``filling in'' subcubes of the graph. These results are applied to compute $H_2(G_n^{sph})$ for the Greene sphere graphs $G^{sph}_n$.

math.AT↗
arXiv2026

Rigidity of multisections to the universal cubic plane curve

We study the following question of Farb: For which positive integers n is it possible to choose n distinct points continuously on every smooth cubic plane curve? This paper is a continuation of the work of Banerjee-Chen. We prove that such choice is unique up to homotopy for n=27 (smallest open case), and does not exist for n=36k+18.

math.AT↗