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Relative homology

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We study the homology of a cell complex $K$ relative to a subcomplex $A$ is, roughly, the homology of $K$ with $A$ "collapsed" to a point.

We form the short exact sequence of the pair: $$0\to C_* (A) \to C_ *(K)\to C_*(K) /C_*(A) \to 0,$$ where $C_*(K)$ is the chain complex of $K$.

The boundary map of $C_*(K)$ does not leave $C_*(A)$ and therefore allows one to define the boundary operator on the quotient on the right. This way we form a new chain complex.

Then the relative homology is defined as: $$H_n(K,A) = H_n (C_*(K) /C_*(A)).$$ The relative homology consists of the relative cycles.

The Snake Lemma yields the following long exact sequence: $$\cdots \to H_n(A) \to H_n(K) \to H_n (K,A) \stackrel{\delta}{\to} H_{n-1}(A) \to \cdots ,$$ where the connecting map $δ$ takes a relative cycle, representing a homology class in $H _n (K,A)$, to its boundary, which is a cycle in $A$.

Using the long exact sequence of pairs and the Excision Theorem, one can show that $H _n (X, A)$ is the same as the $n$-th reduced homology groups of the quotient complex $K/A$: $$\tilde H_n(K/A) = H_n(K,A).$$

One can use this fact to compute the homology of spheres of all dimensions.