Gunnar Carlsson, 2009

Definitions

Homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces. Homology itself was developed as a way to analyse and classify manifolds according to their cycles – closed loops (or more generally submanifolds) that can be drawn on a given n dimensional manifold but not continuously deformed into each other 1.

Connectivity information is information concerning the loops and higher dimensional analogues in a space.

Functoriality is the notion that invariants should be related not just to objects being studied, but also to the maps between those objects.

Two mathematical objects are said to be homotopic if one can be continuously deformed into the other.

The k-th Betti number corresponds to an informal notion of the number of independent k-dimensional surfaces in a k-dimensional vector space.

A simplicial complex structure on a space is an expression of the space as a union of points, intervals, triangles, and higher dimensional analogues.

Summary

The homology of simplicial complexes is algorithmically computable; hence, it is desirable to construct a simplicial complex which computes the homology of an underlying space X. One way to do this is to produce a homotopy equivalence from X to the simplicial complex.

Consider the case where there is a large amount of potentially noisy high-dimensional data. A question that can be asked is whether it is possible to infer the Betti numbers of the underlying space from the noisy data. The Cech complex 2 on a metric space, which is a covering by balls of a fixed radius $\epsilon$, is useful because it can be shown to be homotopy equivalent to the underlying Reimannian manifold. However, the data cannot be assumed to always lie on a manifold. The philosophy of selecting all values of $\epsilon$ at once to compute a summary of homological information is known as persistence.

Interesting points

  • “Topology is exactly that branch of mathematics which deals with qualitative geometric information.”