Topology/Exact Sequences

Exact Sequences Homology Groups → 

An exact sequence is a tool used in Algebraic Topology used to extract information from a sequence of chain groups.

Definition edit

Given a sequence of groups   and homomorphisms


is an exact sequence if   for all  , the sequence can be infinite.

Given an exact sequence of chain groups, with this indexing


we have a chain complex.

Short Exact Sequence edit

Given the special case where we have 3 groups with the following homomorphisms


where   is a one-one homomorphism and   is an onto homomorphism, we have a short exact sequence. Short exact sequences have the property  .

Exercises edit

(under construction)

Exact Sequences Homology Groups →