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

$G_1\stackrel{h_1}{\to }{G}_2\stackrel{h_2}{\to }\cdots \stackrel{h_{n-1}}{\to }{G}_n$

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

Given an exact sequence of chain groups, with this indexing

$\cdots \stackrel{{\partial }_2}{\to }{C}_2\stackrel{{\partial }_1}{\to }{C}_1\stackrel{{\partial }_0}{\to }{C}_0$

we have a chain complex.

Short Exact Sequence

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

$G_1\stackrel{h_1}{\to }{G}_2\stackrel{h_2}{\to }{G}_3$

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)

Template:Navigation