# Homology isomorphism of topological spaces

From Topospaces

*This article defines a property of continuous maps between topological spaces*

## Contents

## Definition

### Symbol-free definition

A continuous map between topological spaces is termed a **homology isomorphism** if the functorially induced maps on all homology groups are isomorphisms.

Equivalently, we can say that the induced map between their singular chain complexes is a quism of complexes.

## Relation with other properties

### Stronger properties

- Homeomorphism
- Homotopy equivalence of topological spaces
- Rational homology isomorphism of topological spaces

## What homology isomorphisms tell us

- Any homology isomorphism of topological spaces also induces an isomorphism on the corresponding cohomology rings (as graded rings).
- If both space are path-connected and simply connected, then any homology isomorphism is also a weak homotopy equivalence of topological spaces -- this follows from Hurewicz theorem applied to a pair.
- Thus if both spaces are path-connected, simply connected CW-spaces then any homology isomorphism is a homotopy equivalence of topological spaces.

## Facts

- The existence of a homology isomorphism is much stronger than having isomorphic homology groups. For instance, it actually implies that the spaces have the same cohomology ring, rather than just the same homology groups; it also implies that if they are simply connected, they are actually homotopy-equivalent.
- Thus properties of a topological space that are invariant upto homology isomorphism, could be much finer than properties that merely depend upon the homology groups or on the cohomology ring.