On the Vietoris-Rips complexes and a cohomology theory for metric spaces.

*(English)*Zbl 0928.55003
Quinn, Frank (ed.), Prospects in topology. Proceedings of a conference in honor of William Browder, Princeton, NJ, USA, March 1994. Princeton, NJ: Princeton University Press. Ann. Math. Stud. 138, 175-188 (1995).

Let \(X\) be a pseudo-metric space with pseudo-distance \(d\) (i.e. \(d\) satisfies the axioms of a distance except that \(d(x,y)=0\) does not imply that \(x=y)\). Let \(\varepsilon\) be a positive real number. We denote by \(X_\varepsilon\) the abstract simplicial complex defined as follows: the vertices of \(X_\varepsilon\) are the points of \(X\) and a \(q\)-simplex of \(X_\varepsilon\) is a subset \(\{x_0, \dots, x_q\}\) of \(X\) such that \(\text{Diam} (\{x_0, \dots, x_q \}) <\varepsilon\). Recall that if \(P\subset X\), the diameter \(\text{Diam} P\) of \(P\) is the least upper bound of \(d(x,y)\) for all \(x,y\in P\).

The complex \(X_\varepsilon\) was once called the Vietoris complex. Since its re-introduction by E. Rips for studying hyperbolic groups, it has been popularized under the name of Rips complex.

This paper is devoted to the study of \(X_\varepsilon\), principally when \(\varepsilon\) is small (in contrast with the framework of Rips). In §2, we establish a few general properties of the complexes \(X_\varepsilon\), the most interesting being that \(| X_\varepsilon |\) is homotopy equivalent to \(|\widehat X_\varepsilon |\) if \(\widehat X\) is the metric completion of \(X\). In §3, we prove theorems concerning the \(\varepsilon\)-complex of a Riemannian manifold (or of a geodesic space). The main consequence of these results is that for \(M\) a compact Riemannian manifold and \(\varepsilon\) small enough, the geometric realization of \(M_\varepsilon\) is homotopy equivalent to \(M\). In §4, we introduce the metric cohomology \({\mathcal H}^*(X;{\mathcal R})\) of a pseudo-metric space \(X\) in a coefficient ring \({\mathcal R}\). The group \({\mathcal H}^* (X;{\mathcal R})\) is defined as the inductive limit of the simplicial cohomology \(H^* (X_\varepsilon; {\mathcal R})\) when \(\varepsilon\to 0\). We show that this produces a cohomology theory for the category of pseudo-metric spaces and uniformly continuous maps (with a “metric version” of the excision axiom and Mayer-Vietoris sequences). In §5, a short direct proof is given that if \(X\) is a compact metric space, then \({\mathcal H}^* (X;{\mathcal R})\) is canonically isomorphic to the Čech cohomology \(\check H^*(X;{\mathcal R})\). We finish by other properties of the metric cohomology which are quite different than those of previously known cohomology theories: for instance, the metric cohomology of \(X\) is isomorphic to that of its metric completion. Also, a contractible metric space might not be acyclic for the metric cohomology.

The corresponding construction for homology \(({\mathcal H}_* (X)\) being the projective limit of \(=H_* (X_\varepsilon))\) was the one originally considered by L. Vietoris (for compact metric spaces) as one of the first attempts to define homology groups for spaces which were not triangulated. Thus, “Vietoris cohomology” could have been an appropriate denomination for our cohomology. We think however that “metric cohomology” is less confusing, since the definition of Vietoris homology has been subsequently modified in the literature in such a way to become the homology counterpart of the Alexander-Spanier cohomology (which is not isomorphic to the metric cohomology).

For the entire collection see [Zbl 0833.00037].

The complex \(X_\varepsilon\) was once called the Vietoris complex. Since its re-introduction by E. Rips for studying hyperbolic groups, it has been popularized under the name of Rips complex.

This paper is devoted to the study of \(X_\varepsilon\), principally when \(\varepsilon\) is small (in contrast with the framework of Rips). In §2, we establish a few general properties of the complexes \(X_\varepsilon\), the most interesting being that \(| X_\varepsilon |\) is homotopy equivalent to \(|\widehat X_\varepsilon |\) if \(\widehat X\) is the metric completion of \(X\). In §3, we prove theorems concerning the \(\varepsilon\)-complex of a Riemannian manifold (or of a geodesic space). The main consequence of these results is that for \(M\) a compact Riemannian manifold and \(\varepsilon\) small enough, the geometric realization of \(M_\varepsilon\) is homotopy equivalent to \(M\). In §4, we introduce the metric cohomology \({\mathcal H}^*(X;{\mathcal R})\) of a pseudo-metric space \(X\) in a coefficient ring \({\mathcal R}\). The group \({\mathcal H}^* (X;{\mathcal R})\) is defined as the inductive limit of the simplicial cohomology \(H^* (X_\varepsilon; {\mathcal R})\) when \(\varepsilon\to 0\). We show that this produces a cohomology theory for the category of pseudo-metric spaces and uniformly continuous maps (with a “metric version” of the excision axiom and Mayer-Vietoris sequences). In §5, a short direct proof is given that if \(X\) is a compact metric space, then \({\mathcal H}^* (X;{\mathcal R})\) is canonically isomorphic to the Čech cohomology \(\check H^*(X;{\mathcal R})\). We finish by other properties of the metric cohomology which are quite different than those of previously known cohomology theories: for instance, the metric cohomology of \(X\) is isomorphic to that of its metric completion. Also, a contractible metric space might not be acyclic for the metric cohomology.

The corresponding construction for homology \(({\mathcal H}_* (X)\) being the projective limit of \(=H_* (X_\varepsilon))\) was the one originally considered by L. Vietoris (for compact metric spaces) as one of the first attempts to define homology groups for spaces which were not triangulated. Thus, “Vietoris cohomology” could have been an appropriate denomination for our cohomology. We think however that “metric cohomology” is less confusing, since the definition of Vietoris homology has been subsequently modified in the literature in such a way to become the homology counterpart of the Alexander-Spanier cohomology (which is not isomorphic to the metric cohomology).

For the entire collection see [Zbl 0833.00037].

##### MSC:

55N35 | Other homology theories in algebraic topology |

54E25 | Semimetric spaces |

54E99 | Topological spaces with richer structures |