路径同调链的归纳构造与\(Ω_3(G;R)\)的结构
Inductive construction of path homology chains and the structure of $Ω_3(G;R)$
浏览论文内容
中文总结 AI 辅助
本文引入归纳方法，从低二维元素构造路径同调链模\(\Omega_n(G;R)\)的元素，由面多重图参数化。该方法为路径链复形提供新结构，可用于理解路径同调。利用此方法构造\(\Omega_3(G;R)\)的显式生成元，回答了Grigor'yan的开放问题并得到通用系数结论。
中文摘要 AI 辅助
路径同调在图拓扑和GLMY理论中起着核心作用。计算图\(G\)的路径同调是一个两步过程，目前文献中甚至对底层链复形都没有完整描述。本文引入一种归纳方法，从低二维的元素构造路径同调链模\(\Omega_n(G;R)\)的元素，通过上下扩展，由面多重图参数化。当\(R\)特征为\(2\)时，构造的归纳元素生成\(\Omega_*(G;R)\)；特征为\(0\)时，至少生成\(i = 0,1,2,3\)的\(\Omega_i(G;R)\)。低维时归纳元素与自然生成元一致，无多重正方形时与Fu和Ivanov的基元素一致。归纳元素为路径链复形提供新结构，可直接用于理解路径同调。利用归纳元素为特征\(0\)或\(2\)的环\(R\)构造\(\Omega_3(G;R)\)的显式生成元，回答了Grigor'yan的一个开放问题，并得到路径同调的几个通用系数结论。
英文摘要
Path homology plays a central role in digraph topology and GLMY theory more generally. Unfortunately, the computation of the path homology of a digraph $G$ is a two-step process, and until now no complete description of even the underlying chain complex has appeared in the literature. In this paper we introduce an inductive method of constructing elements of the path homology chain modules $Ω_n(G;R)$ from elements in the preceding two dimensions. This proceeds via the formation of what we call upper and lower extensions, that are parametrised by certain labelled multigraphs which we introduce and call face multigraphs. The inductive elements we construct generate $Ω_*(G;R)$ when $R$ has characteristic $2$. With characteristic $0$ coefficients, the inductive elements at least generate $Ω_i(G;R)$ for $i=0,1,2,3$. In low dimensions, the inductive elements coincide with the natural generators, and when the digraph contains no multisquares, the inductive elements coincide with the basis elements produced by Fu and Ivanov. Inductive elements provide a new concrete structure on the path chain complex that can be directly applied to understand path homology, under no restriction on the digraph $G$. We employ inductive elements to construct explicit generators of $Ω_3(G;R)$ for a ring $R$ of characteristic $0$ or $2$, answering an open question posed by Grigor'yan. Several universal coefficient statements for path homology are obtained as a byproduct.
