Homotopy category of chain complexes

In the mathematical subfield of homological algebra a chain homotopy is an abstraction of the topological concept of homotopy.

Definition

Let and Kokettenkomplexe and two chain mappings, ie Systems of morphisms that are compatible with the differentials in the sense that is true.

Then a chain homotopy is a sequence of morphisms, so that, or detailed

Applies.

And called homotopic if there is a chain homotopy. Homotopy is compatible with the composition equivalence relation on the set of chain maps.

Homotopies of maps between chain complexes (and not Kokettenkomplexen ) are defined analogously. Two pictures chain and between chain complexes and are called homotopic if there is a sequence of morphisms, so that

Two chain complexes and hot kettenhomotopieäquivalent if there are chain and images for the sequential execution and each homotopic to the identity.

Importance

• An illustration is homotopic to zero mapping is called, is null homotopic. The category of Kokettenkomplexe modulo nullhomotoper pictures is the homotopy category.

• Homotopic chain maps induce the same map in homology and cohomology.

• Is on a particular Kokettenkomplex and a homotopy between the identity and the zero mapping, so the cohomology of is trivial, ie is exact. We also speak of a contracting homotopy.

• If two continuous maps between topological spaces and homotopic, so are the associated figures and between the corresponding singular chain complexes homotopic in the sense defined above. In particular, the induced Abbdildungen between the singular homology groups are the same.

• Two projective resolutions of a module over a ring are homotopic. In particular, the homologies of the resolutions are the same, which leads to the concept of the derived functor.