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Exact sequences

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A sequence $$\cdots \;\xrightarrow{}\; G_0 \;\xrightarrow{f_1}\; G_1 \;\xrightarrow{f_2}\; G_2 \;\xrightarrow{f_3}\; \cdots \;\xrightarrow{f_n}\; G_n \;\xrightarrow{}\; \cdots$$ of groups and homomorphisms is called exact if the image of each homomorphism is equal to the kernel of the next: $$\mathrm{im}(f_k) = \mathrm{ker}(f_{k+1}).$$ The sequence could also be one of vector spaces and linear operators, etc.

Note that both chain complex and cochain complex are similar to this with $$\mathrm{im}(\partial_k) \subset \mathrm{ker}(\partial_{k+1})$$ and $$\mathrm{im}(d_k) \subset \mathrm{ker}(d_{k+1}),$$ respectively. They just aren't exact!

They aren't exact in general, that is. They are if the spaces are "trivial".

We have seen an exact sequence in relative homology. Indeed, given a complex and a subcomplex $A\subset X$, we have $$0\to C_*(A) \to C_*(X)\to C_*(X) /C_*(A) \to 0,$$ where $C_*(X)$ denotes the group of chains in $X$ and the middle arrows are induced by the inclusion and the quotient map respectively.