Inducing a map on homology from a correspondence

TL;DR

This paper explores how to induce homology maps from correspondences between topological spaces, allowing for accurate homological information retrieval even with incomplete or perturbed data, without requiring acyclic preimages.

Contribution

It introduces conditions under which homology homomorphisms from correspondences match those from continuous maps, broadening applicability beyond classical assumptions.

Findings

Homology homomorphisms can be induced from correspondences without acyclic preimages.

Correct homological information can be retrieved despite data missing or being perturbed.

Application demonstrated on combinatorial maps approximating or reconstructing continuous maps.

Abstract

We study the homomorphism induced in homology by a closed correspondence between topological spaces, using projections from the graph of the correspondence to its domain and codomain. We provide assumptions under which the homomorphism induced by an outer approximation of a continuous map coincides with the homomorphism induced in homology by the map. In contrast to more classical results we do not require that the projection to the domain have acyclic preimages. Moreover, we show that it is possible to retrieve correct homological information from a correspondence even if some data is missing or perturbed. Finally, we describe an application to combinatorial maps that are either outer approximations of continuous maps or reconstructions of such maps from a finite set of data points.