Topology/Exact Sequences

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

Definition

Given a sequence of groups $G_{1},G_{2},\dots ,G_{n}$ and homomorphisms

$G_{1}{\xrightarrow {h_{1}}}G_{2}{\xrightarrow {h_{2}}}\cdots {\xrightarrow {h_{n-1}}}G_{n}$

is an exact sequence if $im(h_{k})=ker(h_{k+1})$ for all $1\leq k<n$, the sequence can be infinite.

Given an exact sequence of chain groups, with this indexing

$\cdots {\xrightarrow {\partial _{2}}}C_{2}{\xrightarrow {\partial _{1}}}C_{1}{\xrightarrow {\partial _{0}}}C_{0}$

we have a chain complex.

Short Exact Sequence

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

$G_{1}{\xrightarrow {h_{1}}}G_{2}{\xrightarrow {h_{2}}}G_{3}$

where $h_{1}$ is a one-one homomorphism and $h_{2}$ is an onto homomorphism, we have a short exact sequence. Short exact sequences have the property $G_{3}\cong G_{2}/h_{1}(G_{1})$.

Exercises

(under construction)