Eilenberg–Zilber theorem

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In mathematics, specifically in algebraic topology, the Eilenberg–Zilber theorem is an important result in establishing the link between the homology groups of a product space and those of the spaces and . The theorem first appeared in a 1953 paper in the American Journal of Mathematics by Samuel Eilenberg and Joseph A. Zilber. One possible route to a proof is the acyclic model theorem.

Statement of the theorem

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The theorem can be formulated as follows. Suppose   and   are topological spaces, Then we have the three chain complexes  ,  , and  . (The argument applies equally to the simplicial or singular chain complexes.) We also have the tensor product complex  , whose differential is, by definition,

 

for   and  ,   the differentials on  , .

Then the theorem says that we have chain maps

 

such that   is the identity and   is chain-homotopic to the identity. Moreover, the maps are natural in   and  . Consequently the two complexes must have the same homology:

 

Statement in terms of composite maps

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The original theorem was proven in terms of acyclic models but more mileage was gotten in a phrasing by Eilenberg and Mac Lane using explicit maps. The standard map   they produce is traditionally referred to as the Alexander–Whitney map and   the Eilenberg–Zilber map. The maps are natural in both   and   and inverse up to homotopy: one has

 

for a homotopy   natural in both   and   such that further, each of  ,  , and   is zero. This is what would come to be known as a contraction or a homotopy retract datum.

The coproduct

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The diagonal map   induces a map of cochain complexes   which, followed by the Alexander–Whitney   yields a coproduct   inducing the standard coproduct on  . With respect to these coproducts on   and  , the map

 ,

also called the Eilenberg–Zilber map, becomes a map of differential graded coalgebras. The composite   itself is not a map of coalgebras.

Statement in cohomology

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The Alexander–Whitney and Eilenberg–Zilber maps dualize (over any choice of commutative coefficient ring   with unity) to a pair of maps

 

which are also homotopy equivalences, as witnessed by the duals of the preceding equations, using the dual homotopy  . The coproduct does not dualize straightforwardly, because dualization does not distribute over tensor products of infinitely-generated modules, but there is a natural injection of differential graded algebras   given by  , the product being taken in the coefficient ring  . This   induces an isomorphism in cohomology, so one does have the zig-zag of differential graded algebra maps

 

inducing a product   in cohomology, known as the cup product, because   and   are isomorphisms. Replacing   with   so the maps all go the same way, one gets the standard cup product on cochains, given explicitly by

 ,

which, since cochain evaluation   vanishes unless  , reduces to the more familiar expression.

Note that if this direct map   of cochain complexes were in fact a map of differential graded algebras, then the cup product would make   a commutative graded algebra, which it is not. This failure of the Alexander–Whitney map to be a coalgebra map is an example the unavailability of commutative cochain-level models for cohomology over fields of nonzero characteristic, and thus is in a way responsible for much of the subtlety and complication in stable homotopy theory.

Generalizations

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An important generalisation to the non-abelian case using crossed complexes is given in the paper by Andrew Tonks below. This give full details of a result on the (simplicial) classifying space of a crossed complex stated but not proved in the paper by Ronald Brown and Philip J. Higgins on classifying spaces.

Consequences

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The Eilenberg–Zilber theorem is a key ingredient in establishing the Künneth theorem, which expresses the homology groups   in terms of   and  . In light of the Eilenberg–Zilber theorem, the content of the Künneth theorem consists in analysing how the homology of the tensor product complex relates to the homologies of the factors.

See also

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References

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  • Eilenberg, Samuel; Zilber, Joseph A. (1953), "On Products of Complexes", American Journal of Mathematics, vol. 75, no. 1, pp. 200–204, doi:10.2307/2372629, JSTOR 2372629, MR 0052767.
  • Hatcher, Allen (2002), Algebraic Topology, Cambridge University Press, ISBN 978-0-521-79540-1.
  • Tonks, Andrew (2003), "On the Eilenberg–Zilber theorem for crossed complexes", Journal of Pure and Applied Algebra, vol. 179, no. 1–2, pp. 199–230, doi:10.1016/S0022-4049(02)00160-3, MR 1958384.
  • Brown, Ronald; Higgins, Philip J. (1991), "The classifying space of a crossed complex", Mathematical Proceedings of the Cambridge Philosophical Society, vol. 110, pp. 95–120, CiteSeerX 10.1.1.145.9813, doi:10.1017/S0305004100070158.