Acyclic Models

Acyclic Models is a powerful technique for constructing maps between chain complexes. This method can be used to show the equivalence of chain complexes, and thereby construct isomorphisms of homology theories for spaces. Four elementary uses of acyclic models are:

Excision
Eilenberg-Zilber theorem
Construction of Derived functors (for example, Ext and Tor)
Commutativity of the Cup Product

The Setup

Let $\mathcal{C}$ be a category, and $\mathcal{M}$ be a collection of objects in $\mathcal{C}$ , called model objects. Let $T : \mathcal{C} \to \mathrm{Ab}$ be a functor, where $\mathrm{Ab}$ is the category of abelian groups. If there are objects $g_M \in T(M)$ for $M \in \mathcal{M}$ such that $T(A)$ is freely generated by the images $T(f)(g_M)$ , then $T$ is said to be free with models $\mathcal{M}$ .

For example, the singular $n$ -chains $C_n(X)$ on a space $X$ are free with a single model $\Delta^n$ .

The Big Theorem

Let $K$ and $L$ be covariant functors on $\mathcal{C}$ with values in the category of chain complexes, and let $f : K \to L$ be a chain map defined in dimensions $< q$ . Then if $K_n$ is free with models $\mathcal{M}$ for all $n \ge q$ and $H_n(L(M)) = 0$ for all $n \ge q-1$ and all $M \in \mathcal{M}$ , then $f$ extends to a chain map $f' : K \to L$ ; furthermore, $f'$ is unique up to a chain homotopy $D$ satisfying $D_n = 0$ for every $n < q$ .

In words: if $K$ is free with models and $L$ is acyclic on those models, then partially defined maps $K \to L$ can be extended uniquely up to chain homotopy. The interesting corollary is that if $K$ and $L$ are both free and acyclic on the same models, then they are chain equivalent, provided an isomorphism in dimension zero.

This can be particularly useful because the reduced singular chain complex of a space is free with models $\Delta^n$ , and since simplices are contractible $\widetilde{H}_*(\Delta^n) = 0.$ Note that the (unreduced) singular chain complex is not acyclic since $H_0(\Delta^n) = \mathbb{Z}$ .

In my next post, I will sketch a proof of the excision theorem in singular homology, using the method of Acyclic Models.

References

Samuel Eilenberg and Saunders MacLane, “Acyclic models,” American Journal of Mathematics, vol. 75 (1953), pp. 189-199

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One Response to “Acyclic Models”

Acyclic Models, the Eilenberg-Zilber Theorem, and Excision « Diagram Chasing Says:
August 13, 2009 at 2:45 pm | Reply
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Acyclic Models

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