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)