Skeletal Homology

Abstract

"Skeletal homology" $K_{n}^{\varepsilon}(X)$ refers to the homology of the chain complex $S_{n}^{\varepsilon}(X)$ generated by skeletal $(n,\varepsilon)$-simplices, i.e. functions from the $0$-skeleton of the standard simplex into a metric space $X$, with image diameter less than $\varepsilon>0$. This homology was previously defined by Goldfarb, who showed that for finite metric spaces, it is isomorphic to the simplicial homology $H_{n}^{\Delta}(VR_{\varepsilon}(X))$ of the VR complex. We prove an isomorphism for arbitrary metric spaces, and introduce new methods to understand homology at scale. We define an invariant metric on $S_{n}^{\varepsilon}(X)$, called the ultradiamond metric, that extends the uniform metric on skeletal simplices. With this metric we prove that "close cycles are homologous", which quickly leads to a host of stability results. We modify methods from singular homology to prove a strong generalization of Hausmann's Theorem, one of the two main justifications to use $H_{n}^{\Delta}(VR_{\varepsilon}(X))$ as a proxy for homology in discrete metric spaces. The second justification is Latchev's Theorem, for which we also prove a strong generalization. We define a homomorphism $\rho_{\varepsilon}:H_{n}(X)\rightarrow K_{n}^{\varepsilon}(X)$ induced by repeated barycentric subdivision and restriction, the image of which we call "real homology" $H_{n}^{\varepsilon}(X)$ at scale. We argue that $H_{n}^{\varepsilon}(X)$ better represents bona fide homology at scale than $H_{n}^{\Delta }(VR_{\varepsilon}(X))$. To distinguish them, we define "phantom homology" to be $P_{n}^{\varepsilon}(X)=K_{n}^{\varepsilon}(X)/H_{n}^{\varepsilon}(X)$, and use the stability of $K_{n}^{\varepsilon}(X)$ to show that in collapse of Riemannian manifolds (e.g. the Berger Spheres), phantom homology can anticipate the abrupt drop in dimension that occurs in the limit.