论文标题
大小同源和路径同源性
Magnitude homology and Path homology
论文作者
论文摘要
在本文中，我们表明同源性和路径同源性密切相关，我们提供了一些应用。我们定义差异$ {\ MATHRM MH}^{\ ell} _K（g）\ longrightArrow {\ Mathrm Mh}^{\ ell-1} _ {k-1} _ {k-1}（g）$介于digraph $ g $的幅度同源之间，这使它们成为链链的复合物。然后，我们证明其同源性$ {\ Mathcal MH}^{\ ell} _k（g）$在Grigor'yanyan-Muranov-s.-t开发的“ digraphs''同型digraphs'的背景下，是非平凡的，同型不变。 Yau等人（下面的G-M -ys）。值得注意的是，我们同源性的对角部分$ {\ MATHCAL MH}^{k} _K（g）$与还原的路径同源性$ \ tilde {h} _k（g）$同构。此外，我们构建了一个频谱序列，其第一页与幅度同源性$ {\ mathrm mh}^{\ ell} _k（g）$同级同级，第二页与我们的同源$ {\ Mathcal MH}^{\ ell}^{\ ell} _K（g）$同构。作为应用，我们表明同源性的幅度同源性的对角性意味着降低路径同源性的微不足道。我们还表明$ \ tilde {h} _k（g）= 0 $ for $ k \ geq 2 $和$ \ tilde {h} _1（g）\ neq 0 $，如果未方向图$ g $的任何边缘包含在长度$ \ geq 5 $的周期中。
In this article, we show that magnitude homology and path homology are closely related, and we give some applications. We define differentials ${\mathrm MH}^{\ell}_k(G) \longrightarrow {\mathrm MH}^{\ell-1}_{k-1}(G)$ between magnitude homologies of a digraph $G$, which make them chain complexes. Then we show that its homology ${\mathcal MH}^{\ell}_k(G)$ is non-trivial and homotopy invariant in the context of `homotopy theory of digraphs' developed by Grigor'yan--Muranov--S.-T. Yau et al (G-M-Ys in the following). It is remarkable that the diagonal part of our homology ${\mathcal MH}^{k}_k(G)$ is isomorphic to the reduced path homology $\tilde{H}_k(G)$ also introduced by G-M-Ys. Further, we construct a spectral sequence whose first page is isomorphic to magnitude homology ${\mathrm MH}^{\ell}_k(G)$, and the second page is isomorphic to our homology ${\mathcal MH}^{\ell}_k(G)$. As an application, we show that the diagonality of magnitude homology implies triviality of reduced path homology. We also show that $\tilde{H}_k(g) = 0$ for $k \geq 2$ and $\tilde{H}_1(g) \neq 0$ if any edges of an undirected graph $g$ is contained in a cycle of length $\geq 5$.
