§ Homology, the big picture

First, there was nothing. Then, we decided we want homology. We start out by baby-stepping with simplicial homology. We rapidly abandon this in favour of singular homology, since it's easier to define notions of chain morphisms with singular homology. We want to know when spaces have the same homology/chain complex. A reasonable idea is to prove that when the map between spaces is homotopic to identity, the chain complex is identical. During the course of the proof, we feel the need for a notion of "chain equivalence", and thus we we are led to define the notion of chain homotopy , and more generally, the homotopy category of chain complexes. Next, we want the long exact sequence of homology connecting a space to its quotient. To get here, we first consider relative homology, connecting a space to a subspace. Next, we are led to build the machinery of excision which tells us that relative homlogy is unchanged on removing a subspace. Finally, we craft "good pairs", which are spaces for whom the relative homology sequence will be equivalent to the quotient long exact sequence. Excision lets us prove that relative homology is equal to quotient homology for good pairs. Once we have excision, we use it to show that show that (1) simplicial and singular homology agree, and (2) construct the Mayer Viteoris sequence (the Kunneth of homology).

§ Chain equivalence

Two chains C,DC, D are chain equivalent if there's a prism operator betwen them :) More seriously,

§ Relative homology

Build sequence Cn(X)Cn(A)Cn(X)/Cn(A)C_n(X) \rightarrow C_n(A) \rightarrow C_n(X)/C_n(A). Snake lemma gives us long exact sequence of relative homology.

§ Good pair

(X,A)(X, A) is a good pair if (1) AA is closed in XX (contains all its boundaries), (2) there is an open nbhd UU of AA ( AUXA \subseteq U \subseteq X) such that UU deformation retracts onto AA. For example, (D2,S1)(D^2, S^1) is a good pair, because the subspace U=D2(0,0)U = D^2 - (0, 0) (annulus) deformation retracts onto the boundary S1S^1. The subspace UU is open since it is the complement of a closed set, the point. More generally, given an attaching map f:D2Xf: \partial D^2 \rightarrow X to attach a D2D^2 into a space XX, if we build YXfD2Y \equiv X \cup_f D^2, then (Y,X)(Y, X) is a good pair, since we have an open UU which is YY minus the origin of the disk that conatins XX. That is, set UXf(D2(0,0))U \equiv X \cup_f (D^2 - (0, 0)). UU contains XX, and deformation retracts onto XX, since we can "deform" the ball minus the origin D2(0,0)D^2 - (0, 0) back into the original space XX. This, cellular attachments create a "chain of good spaces". Also see this answer on HEP versus good pair

§ good pair + Excision => relative equals quotient homology

Let (X,A)(X, A) be a good pair. Consider the projection map which kills AA, the map π:(X,A)(X/A,[A])\pi: (X, A) \rightarrow (X/A, [A]) as a map of good pairs. (Note that (X/A,[A])(X/A, [A]) is also a good pair). We claim this induces isomorphisms π:Hn(X,A)Hn(X/A,[A])\pi^*: H_n(X, A) \rightarrow H_n(X/A, [A]).