Long Exact Sequence of a Pair

Algebraic Topology · Axiom Academy

LESSON Long Exact Sequence of a Pair Connecting homology of subspaces, spaces, and relative homology Step 1: What is a Pair in Algebraic Topology? In algebraic topology, we often want to understand the structure of a space in relation to one of its subspaces. The concept of a pair formalizes this relationship and allows us to measure the "difference" between a space and its subspace using homology. A pair in algebraic topology is an ordered pair where X is a topological space and A is a subspace of X (i.e., ). The subspace A is often called the base space or base pair , and the pair (X, A) allows us to study how the homology of X relates to that of A . Think of (X, A) as a "space with a distinguished subspace." For example: Disk and its boundary: (D^n, S^ n-1 ) – a closed ball and its boundary sphere Sphere and a point: (S^1, *) – a circle with a marked point Manifold and submanifold: (M, N) – a manifold containing a submanifold The fundamental idea is to define homology groups that measure the difference between the chain complex of X and the chain complex of A . Given a pair (X, A) , the relative chain complex is defined as: In other words, relative chains are cosets of chains in A modulo chains in X . The relative homology groups are then defined as:

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