Computability of Homology for Compact Absolute Neighbourhood Retracts

In this note we discuss the information needed to compute the homology groups of a topological space. We argue that the natural class of spaces to consider are the compact absolute neighbourhood retracts, since for these spaces the homology groups are finite. We show that we need to specify both a function which defines a retraction from a neighbourhood of the space in the Hilbert cube to the space itself, and a sufficiently fine over-approximation of the set. However, neither the retraction itself, nor a description of an approximation of the set in the Hausdorff metric, is sufficient to compute the homology groups. We express the conditions in the language of computable analysis, which is a powerful framework for studying computability in topology and geometry, and use cubical homology to perform the computations.