In algebraic topology, a discipline within mathematics, the acyclic models theorem can be used to show that two homology theories are isomorphic. The theorem was developed by topologists Samuel Eilenberg and Saunders MacLane.[1] They discovered that, when topologists were writing proofs to establish equivalence of various homology theories, there were numerous similarities in the processes. Eilenberg and MacLane then discovered the theorem to generalize this process.

It can be used to prove the Eilenberg–Zilber theorem; this leads to the idea of the model category.

Statement of the theorem

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Let   be an arbitrary category and   be the category of chain complexes of  -modules over some ring  . Let   be covariant functors such that:

  •   for  .
  • There are   for   such that   has a basis in  , so   is a free functor.
  •   is  - and  -acyclic at these models, which means that   for all   and all  .

Then the following assertions hold:[2][3]

  • Every natural transformation   induces a natural chain map  .
  • If   are natural transformations,   are natural chain maps as before and   for all models  , then there is a natural chain homotopy between   and  .
  • In particular the chain map   is unique up to natural chain homotopy.

Generalizations

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Projective and acyclic complexes

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What is above is one of the earliest versions of the theorem. Another version is the one that says that if   is a complex of projectives in an abelian category and   is an acyclic complex in that category, then any map   extends to a chain map  , unique up to homotopy.

This specializes almost to the above theorem if one uses the functor category   as the abelian category. Free functors are projective objects in that category. The morphisms in the functor category are natural transformations, so the constructed chain maps and homotopies are all natural. The difference is that in the above version,   being acyclic is a stronger assumption than being acyclic only at certain objects.

On the other hand, the above version almost implies this version by letting   a category with only one object. Then the free functor   is basically just a free (and hence projective) module.   being acyclic at the models (there is only one) means nothing else than that the complex   is acyclic.

Acyclic classes

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There is a grand theorem that unifies both of the above.[4][5] Let   be an abelian category (for example,   or  ). A class   of chain complexes over   will be called an acyclic class provided that:

  • The 0 complex is in  .
  • The complex   belongs to   if and only if the suspension of   does.
  • If the complexes   and   are homotopic and  , then  .
  • Every complex in   is acyclic.
  • If   is a double complex, all of whose rows are in  , then the total complex of   belongs to  .

There are three natural examples of acyclic classes, although doubtless others exist. The first is that of homotopy contractible complexes. The second is that of acyclic complexes. In functor categories (e.g. the category of all functors from topological spaces to abelian groups), there is a class of complexes that are contractible on each object, but where the contractions might not be given by natural transformations. Another example is again in functor categories but this time the complexes are acyclic only at certain objects.

Let   denote the class of chain maps between complexes whose mapping cone belongs to  . Although   does not necessarily have a calculus of either right or left fractions, it has weaker properties of having homotopy classes of both left and right fractions that permit forming the class   gotten by inverting the arrows in  .[4]

Let   be an augmented endofunctor on  , meaning there is given a natural transformation   (the identity functor on  ). We say that the chain complex   is  -presentable if for each  , the chain complex

 

belongs to  . The boundary operator is given by

 .

We say that the chain complex functor   is  -acyclic if the augmented chain complex   belongs to  .

Theorem. Let   be an acyclic class and   the corresponding class of arrows in the category of chain complexes. Suppose that   is  -presentable and   is  -acyclic. Then any natural transformation   extends, in the category   to a natural transformation of chain functors   and this is unique in   up to chain homotopies. If we suppose, in addition, that   is  -presentable, that   is  -acyclic, and that   is an isomorphism, then   is homotopy equivalence.

Example

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Here is an example of this last theorem in action. Let   be the category of triangulable spaces and   be the category of abelian group valued functors on  . Let   be the singular chain complex functor and   be the simplicial chain complex functor. Let   be the functor that assigns to each space   the space

 .

Here,   is the  -simplex and this functor assigns to   the sum of as many copies of each  -simplex as there are maps  . Then let   be defined by  . There is an obvious augmentation   and this induces one on  . It can be shown that both   and   are both  -presentable and  -acyclic (the proof that   is presentable and acyclic is not entirely straightforward and uses a detour through simplicial subdivision, which can also be handled using the above theorem). The class   is the class of homology equivalences. It is rather obvious that   and so we conclude that singular and simplicial homology are isomorphic on  .

There are many other examples in both algebra and topology, some of which are described in [4][5]

References

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  1. ^ S. Eilenberg and S. Mac Lane (1953), "Acyclic Models." Amer. J. Math. 75, pp.189–199
  2. ^ Joseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag ISBN 0-387-96678-1 (See chapter 9, thm 9.12)
  3. ^ Dold, Albrecht (1980), Lectures on Algebraic Topology, A Series of Comprehensive Studies in Mathematics, vol. 200 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 3-540-10369-4
  4. ^ a b c M. Barr, "Acyclic Models" (1999).
  5. ^ a b M. Barr, Acyclic Models (2002) CRM monograph 17, American Mathematical Society ISBN 978-0821828779.
  • Schon, R. "Acyclic models and excision." Proc. Amer. Math. Soc. 59(1) (1976) pp.167--168.