Relative Homology Examples

Algebraic Topology · Axiom Academy

EXAMPLE Relative Homology Examples Computing H_n(D^n, S^(n-1)) and understanding relative homology through exact sequences Excellent work! You've mastered relative homology calculations. Here's what we learned: Long Exact Sequence: The sequence of a pair (X, A) relates H_*(X), H_*(A), and H_*(X, A) through a powerful exact sequence that enables computations. Relative = Quotient: H_n(D^n, S^(n-1)) measures cycles in D^n that are not already in S^(n-1). The n-disk modulo its boundary is essentially S^n. Good Pairs: For a "good pair" (D^n, S^(n-1)), where the subspace is contained in the interior's closure, relative homology behaves particularly nicely. Excision Power: Combined with excision, relative homology lets us compute H_*(S^n) by removing a point and using H_n(D^n, S^(n-1)) ≅ Z. Relative homology is essential for understanding quotient spaces, attaching cells in CW complexes, and proving excision theorems!

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