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.

Definition[edit | edit source]

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 | edit source]

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 | edit source]

(under construction)

Exact Sequences Homology Groups →