Trends in Topological Combinatorics

拓扑组合学中的图子式定理

The graph minor theorem in topological combinatorics

本文从图子式范畴视角研究匹配复形等拓扑组合构造，证明有界亏格图的同调挠部分一致有界，给出了Robertson-Seymour图子式定理的代数版本。

Cohen-Macaulay and Gorenstein complexes from a topological point of view

The main invariant to study the combinatorics of a simplicial complex $K$ is the associated face ring or Stanley-Reisner algebra. Reisner respectively Stanley explained in which sense Cohen-Macaulay and Gorenstein properties of the face ring are reflected by geometric and/or combinatoric properties of the simplicial complex. We give a new proof for these result by homotopy theoretic methods and constructions. Our approach is based on ideas used very successfully in the analysis of the homotopy theory of classifying spaces.

Hell-Nešetřil二分法的拓扑证明

A topological proof of the Hell-Nešetřil dichotomy

该研究结合拓扑组合学与约束满足问题的代数方法，基于Lovász的思路给出了图同态问题二分性结论Hell-Nešetřil定理的全新证明。

A user's guide to the topological Tverberg conjecture

The topological Tverberg conjecture was considered a central unsolved problem of topological combinatorics. The conjecture asserts that for any integers $r,d>1$ and any continuous map $f:Δ\to\mathbb R^d$ of the $(d+1)(r-1)$-dimensional simplex there are pairwise disjoint faces $σ_1,\ldots,σ_r\subsetΔ$ such that $f(σ_1)\cap \ldots \cap f(σ_r)\ne\emptyset$. The conjecture was proved for a prime power $r$. Recently counterexamples for other $r$ were found. Analogously, the $r$-fold van Kampen-Flores conjecture holds for a prime power $r$ but does not hold for other $r$. The arguments form a beaut…

拓扑学基本定义中隐含的有限组合学

Finite combinatorics implicit in the basic definitions of topology

本文用基于 Quillen 提升性质的组合记号，将拓扑学中连通性、紧致性等基本定义压缩为极短形式，揭示其隐含的有限前序组合结构。