Long Exact Sequences

Algebraic Topology · Axiom Academy

Connecting homomorphisms, snake lemma idea 1. From Short to Long Exact Sequences While short exact sequences have length 5, exact sequences can continue indefinitely: The most important long exact sequences have connecting homomorphisms that change dimension: 3. The Snake Lemma (Intuition) The snake lemma shows how to construct connecting homomorphisms from a commutative diagram: 4. Long Exact Sequence of a Pair The most important example in algebraic topology: the long exact sequence of a pair (X, A): Long exact sequences extend infinitely Connecting homomorphisms link different dimensions Snake lemma provides the construction Exactness gives computational power

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