论文标题
半圆形空间，同型和同源性
Semi-coarse Spaces, Homotopy and Homology
论文作者
论文摘要
我们开始研究半圆形空间的代数拓扑，这些空间是粗空间的概括，使人们能够赋予非平凡的“粗样”结构以紧凑的度量空间，这在粗几何形状中是不可能的。我们在这种情况下首先研究同型，然后我们构建同源组，这些组在半差异的同性恋等效性下是不变的。我们进一步表明，任何无方向的图$ g =（v，e）$都会在其一组顶点$ v_g $上诱导一个半结构的结构，并且相应的半蛋白同源性与越野式 - 里普斯同源性是同构的。反过来，这导致了无方向图的越野杆同源性的同源性定理。
We begin the study the algebraic topology of semi-coarse spaces, which are generalizations of coarse spaces that enable one to endow non-trivial `coarse-like' structures to compact metric spaces, something which is impossible in coarse geometry. We first study homotopy in this context, and we then construct homology groups which are invariant under semi-coarse homotopy equivalence. We further show that any undirected graph $G=(V,E)$ induces a semi-coarse structure on its set of vertices $V_G$, and that the respective semi-coarse homology is isomorphic to the Vietoris-Rips homology. This, in turn, leads to a homotopy invariance theorem for the Vietoris-Rips homology of undirected graphs.
