Homology isomorphism of topological spaces

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This article defines a property of continuous maps between topological spaces

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

What homology isomorphisms tell us

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.