Exact Sequences Summary

Algebraic Topology · Axiom Academy

SUMMARY Exact Sequences and Excision Key concepts from Unit 6: exact sequences, relative homology, excision, and Mayer-Vietoris. Definition: A sequence 0 A B C 0 where (f) = (g) , and f is injective, g is surjective Splitting: A short exact sequence splits if there exists a homomorphism making it split (i.e., B A C ) Key Property: Exactness means "image equals kernel" at each position Why It Matters: Short exact sequences encode how spaces decompose and relate to quotients Structure: For a pair (X, A) , we get H_n(A) H_n(X) H_n(X, A) H_ n-1 (A) Connecting Homomorphism: The boundary map connects relative to absolute homology Exactness: At each position, image equals kernel, encoding relationships between spaces Applications: Essential for computing homology of quotient spaces and spheres Relative Homology Interpretation Definition: H_n(X, A) measures cycles in X that become boundaries when restricted to A Quotient Perspective: H_n(X, A) _n(X/A) when A is a good subspace Intuition: Captures "what's new" in X beyond what's already in A Computation: Often easier to compute relative homology than absolute Statement: If (A) , then H_n(X - Z, A - Z) H_n(X, A) Meaning: We can "cut out" a piece Z without changing relative homology Conditions: Works when Z sits deeply inside A (closure in interior) Power: Allows decomposition of spaces into simpler pieces for computation Using the Mayer-Vietoris Sequence

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