A sequence of abelian group homomorphisms ... -> A -> B -> C -> ... is exact at B if the image of the incoming map equals the kernel of the outgoing map: im(A -> B) = ker(B -> C). Exact sequences encode relationships between groups with perfect precision — no information is lost or created at any stage. Short exact sequences 0 -> A -> B -> C -> 0 express B as an "extension" of C by A, and long exact sequences are the workhorses of homological computation, connecting the homology of related spaces through an infinite chain of exactness conditions.
An exact sequence is a sequence of abelian groups and homomorphisms ... -> A_{n+1} -f_{n+1}-> A_n -f_n-> A_{n-1} -> ... where the image of each map equals the kernel of the next: im(f_{n+1}) = ker(f_n) for all n. This is a stronger condition than being a chain complex (where im subset ker); exactness means the homology H_n = ker(f_n)/im(f_{n+1}) is zero at every term. Exact sequences encode the tightest possible algebraic relationships between groups.
The most important type is the short exact sequence (SES): 0 -> A -i-> B -p-> C -> 0. Exactness at A says i is injective (A embeds in B). Exactness at C says p is surjective (every element of C is in the image). Exactness at B says im(i) = ker(p) (the copy of A inside B is exactly what gets killed by p). Together: C = B/i(A), and B is an "extension" of C by A. The group B is assembled from A and C, but the SES does not uniquely determine B — there can be multiple non-isomorphic extensions (classified by the Ext functor). The SES splits if B = A direct sum C, which happens when there exists a section s : C -> B with p compose s = id, or equivalently a retraction r : B -> A with r compose i = id.
Long exact sequences (LES) arise naturally in algebraic topology whenever we have a short exact sequence of chain complexes. Given 0 -> A_* -> B_* -> C_* -> 0 (an SES of chain complexes), there is a long exact sequence in homology: ... -> H_n(A) -> H_n(B) -> H_n(C) -partial-> H_{n-1}(A) -> H_{n-1}(B) -> ... The maps H_n(A) -> H_n(B) -> H_n(C) are induced by the chain maps. The connecting homomorphism partial : H_n(C) -> H_{n-1}(A) is the key new ingredient — it does not come from a chain map but is constructed by diagram chasing (the snake lemma). This connecting homomorphism is what links the homology groups across dimensions.
In algebraic topology, long exact sequences appear everywhere. The LES of a pair (X, A) comes from the SES of chain complexes 0 -> C_*(A) -> C_*(X) -> C_*(X, A) -> 0. The Mayer-Vietoris sequence for X = A union B is a long exact sequence derived (via excision) from a similar SES. The LES of a fibration F -> E -> B in homotopy theory comes from the fiber sequence structure. In each case, the long exact sequence provides the computational framework: knowing two of the three families of groups determines the third, up to extension problems that the connecting homomorphisms resolve.
The language of exact sequences pervades all of homological algebra and algebraic topology. A map being injective is equivalent to 0 -> A -> B being exact. A map being surjective is equivalent to B -> C -> 0 being exact. An isomorphism is equivalent to 0 -> A -> B -> 0 being exact. The five lemma and snake lemma are tools for manipulating exact sequences, and the entire framework of derived functors (Ext, Tor, sheaf cohomology) is built on analyzing when functors fail to preserve exactness. Understanding exact sequences is understanding the grammar of homological algebra.