Topology/Exact Sequences

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

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

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

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

Given an exact sequence of chain groups, with this indexing

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

we have a chain complex.

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

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

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

(under construction)