The acyclic models theorem, in the full generality that I stated in my previous post, is not usually the form in which it is used. More generally:

**Corollary** **1**: If and are functors from a category to the category of augmented chain complexes, such that and are free and acyclic on models , then there exists a natural transformation extending the identity map , which is unique up to natural chain homotopy.

**Corollary 2: **If and are functors to the category of (non-augmented) chain complexes which are free, and acyclic in dimensions above zero, then any natural chain map between them that restricts to an isomorphism on is a chain homotopy equivalence.

Corollaries 1 and 2 above are effectively restatements of the same powerful idea. The acyclic models theorem reduces equality arguments in homology to proving that appropriate chain complexes are free and acyclic.

**Theorem (Eilenberg-Zilber):** Let denote the singular chain complex of the space . Then the functors and from to are naturally chain homotopic. As a result, .

*Proof:* By definition both functors and are free with models (in fact, is free with models ). Since the space is contractible, is acyclic in positive dimensions. Since is itself contractible, is chain homotopic to , which is acyclic in positive dimensions as well. Therefore both functors are free and acyclic.

Consider the (natural) map taking . This map is invertible and so induces an isomorphism on . Therefore the two functors are chain homotopic. QED.

And it is, remarkably, that simple, although I spent quite a while convincing myself that I hadn’t missed any steps (and I still hope I haven’t). It just turns out that the acyclic models theorem is incredibly powerful; especially in the proof of the excision theorem.

And, yes, I have done some weasely magic by assuming that homology is invariant over homotopic spaces; although that’s a standard result, the Eilenberg-Zilber theorem does yield a *very* simple proof that homotopic maps induce isomorphisms on homology — one could hypothetically just compute the homology of simplices directly if one wanted to be a purist.

The excision theorem is proven in a similar way; at the core of excision is the claim that if and are subspaces of whose interiors cover , then and are chain homotopic, where is the chain complex of simplices whose image lies completely in or completely in .

Excision is typically proven through *barycentric subdivision*, with the geometric intuition being that every simplex in can be subdivided into smaller simplices, and eventually each simplex will be small enough to be covered either by the interior of or the interior of . However, you could check in Hatcher’s book to see that this argument is far less simple than you would like it to be, taking up a few pages to hammer out all of the algebraic (and geometric!) details.

Instead, the acyclic models theorem opens up another avenue of attack: it is painfully clear that and are isomorphic, since the interiors of and form an open cover of . One can consider the category of *spaces with open coverings*, which has objects , where is an open cover of , and whose maps respect these coverings in the natural way.

Then we can associate to the chain complex of simplices whose image is contained wholly in one of the open sets of , and we can also associate to it the standard chain complex . One can show both are acyclic and free with models , where is the -cube, without too much trouble (I will omit the details, but it’s a basic lebesgue number /compactness argument). Therefore, these functors are naturally chain homotopic, and excision is proved.

(Covering bases: the well-definedness of derived functors is not an acyclic models argument so much as it is nearly identical to the *proof* of acyclic models. Sorry about that. )

Next I will be moving into a brief look at cohomology, cup products, and the power of acyclic models in that context for proving things like associativity and commutativity.

**References**:

- R. Schon,
*Acyclic Models and Excision*. Proceedings of the American Mathematical Society, 1976.