Topology/Exact Sequences

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Exact Sequences Homology Groups → 

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

DefinitionEdit

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 SequenceEdit

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  .

ExercisesEdit

(under construction)


Topology
Exact Sequences Homology Groups →