Eilenberg–Zilber theorem
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 [math]\displaystyle{ X \times Y }[/math] and those of the spaces [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math]. 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
The theorem can be formulated as follows. Suppose [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are topological spaces, Then we have the three chain complexes [math]\displaystyle{ C_*(X) }[/math], [math]\displaystyle{ C_*(Y) }[/math], and [math]\displaystyle{ C_*(X \times Y) }[/math]. (The argument applies equally to the simplicial or singular chain complexes.) We also have the tensor product complex [math]\displaystyle{ C_*(X) \otimes C_*(Y) }[/math], whose differential is, by definition,
- [math]\displaystyle{ \partial_{C_*(X) \otimes C_*(Y)}( \sigma \otimes \tau) = \partial_X \sigma \otimes \tau + (-1)^p \sigma \otimes \partial_Y\tau }[/math]
for [math]\displaystyle{ \sigma \in C_p(X) }[/math] and [math]\displaystyle{ \partial_X }[/math], [math]\displaystyle{ \partial_Y }[/math] the differentials on [math]\displaystyle{ C_*(X) }[/math],[math]\displaystyle{ C_*(Y) }[/math].
Then the theorem says that we have chain maps
- [math]\displaystyle{ F\colon C_*(X \times Y) \rightarrow C_*(X) \otimes C_*(Y), \quad G\colon C_*(X) \otimes C_*(Y) \rightarrow C_*(X \times Y) }[/math]
such that [math]\displaystyle{ FG }[/math] is the identity and [math]\displaystyle{ GF }[/math] is chain-homotopic to the identity. Moreover, the maps are natural in [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math]. Consequently the two complexes must have the same homology:
- [math]\displaystyle{ H_*(C_*(X \times Y)) \cong H_*(C_*(X) \otimes C_*(Y)). }[/math]
Statement in terms of composite maps
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 [math]\displaystyle{ F }[/math] they produce is traditionally referred to as the Alexander–Whitney map and [math]\displaystyle{ G }[/math] the Eilenberg–Zilber map. The maps are natural in both [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] and inverse up to homotopy: one has
- [math]\displaystyle{ FG = \mathrm{id}_{C_*(X) \otimes C_*(Y)}, \qquad GF - \mathrm{id}_{C_*(X \times Y)} = \partial_{C_*(X) \otimes C_*(Y)}H+H\partial_{C_*(X) \otimes C_*(Y)} }[/math]
for a homotopy [math]\displaystyle{ H }[/math] natural in both [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] such that further, each of [math]\displaystyle{ HH }[/math], [math]\displaystyle{ FH }[/math], and [math]\displaystyle{ HG }[/math] is zero. This is what would come to be known as a contraction or a homotopy retract datum.
The coproduct
The diagonal map [math]\displaystyle{ \Delta\colon X \to X \times X }[/math] induces a map of cochain complexes [math]\displaystyle{ C_*(X) \to C_*(X \times X) }[/math] which, followed by the Alexander–Whitney [math]\displaystyle{ F }[/math] yields a coproduct [math]\displaystyle{ C_*(X) \to C_*(X) \otimes C_*(X) }[/math] inducing the standard coproduct on [math]\displaystyle{ H_*(X) }[/math]. With respect to these coproducts on [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math], the map
- [math]\displaystyle{ H_*(X) \otimes H_*(Y) \to H_*\big(C_*(X) \otimes C_*(Y)\big)\ \overset\sim\to\ H_*(X \times Y) }[/math],
also called the Eilenberg–Zilber map, becomes a map of differential graded coalgebras. The composite [math]\displaystyle{ C_*(X) \to C_*(X) \otimes C_*(X) }[/math] itself is not a map of coalgebras.
Statement in cohomology
The Alexander–Whitney and Eilenberg–Zilber maps dualize (over any choice of commutative coefficient ring [math]\displaystyle{ k }[/math] with unity) to a pair of maps
- [math]\displaystyle{ G^*\colon C^*(X \times Y) \rightarrow \big(C_*(X) \otimes C_*(Y)\big)^*, \quad F^*\colon \big(C_*(X) \otimes C_*(Y)\big)^*\rightarrow C^*(X \times Y) }[/math]
which are also homotopy equivalences, as witnessed by the duals of the preceding equations, using the dual homotopy [math]\displaystyle{ H^* }[/math]. 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 [math]\displaystyle{ i\colon C^*(X) \otimes C^*(Y) \to \big(C_*(X) \otimes C_*(Y)\big)^* }[/math] given by [math]\displaystyle{ \alpha \otimes \beta \mapsto (\sigma \otimes \tau \mapsto \alpha(\sigma)\beta(\tau)) }[/math], the product being taken in the coefficient ring [math]\displaystyle{ k }[/math]. This [math]\displaystyle{ i }[/math] induces an isomorphism in cohomology, so one does have the zig-zag of differential graded algebra maps
- [math]\displaystyle{ C^*(X) \otimes C^*(X)\ \overset{i}{\to}\ \big(C_*(X) \otimes C_*(X)\big)^*\ \overset{G^*}{\leftarrow}\ C^*(X \times X) \overset{C^*(\Delta)}{\to} C^*(X) }[/math]
inducing a product [math]\displaystyle{ \smile\colon H^*(X) \otimes H^*(X) \to H^*(X) }[/math] in cohomology, known as the cup product, because [math]\displaystyle{ H^*(i) }[/math] and [math]\displaystyle{ H^*(G) }[/math] are isomorphisms. Replacing [math]\displaystyle{ G^* }[/math] with [math]\displaystyle{ F^* }[/math] so the maps all go the same way, one gets the standard cup product on cochains, given explicitly by
- [math]\displaystyle{ \alpha \otimes \beta \mapsto \Big(\sigma \mapsto (\alpha \otimes \beta)(F^*\Delta^*\sigma) = \sum_{p=0}^{\dim \sigma} \alpha(\sigma|_{\Delta^{[0,p]}}) \cdot \beta(\sigma|_{\Delta^{[p,\dim \sigma]}})\Big) }[/math],
which, since cochain evaluation [math]\displaystyle{ C^p(X) \otimes C_q(X) \to k }[/math] vanishes unless [math]\displaystyle{ p=q }[/math], reduces to the more familiar expression.
Note that if this direct map [math]\displaystyle{ C^*(X) \otimes C^*(X) \to C^*(X) }[/math] of cochain complexes were in fact a map of differential graded algebras, then the cup product would make [math]\displaystyle{ C^*(X) }[/math] 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
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
The Eilenberg–Zilber theorem is a key ingredient in establishing the Künneth theorem, which expresses the homology groups [math]\displaystyle{ H_*(X \times Y) }[/math] in terms of [math]\displaystyle{ H_*(X) }[/math] and [math]\displaystyle{ H_*(Y) }[/math]. 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
References
- Eilenberg, Samuel; Zilber, Joseph A. (1953), "On Products of Complexes", American Journal of Mathematics 75 (1): 200–204, doi:10.2307/2372629.
- 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 179 (1–2): 199–230, doi:10.1016/S0022-4049(02)00160-3.
- Brown, Ronald; Higgins, Philip J. (1991), "The classifying space of a crossed complex", Mathematical Proceedings of the Cambridge Philosophical Society 110: 95–120, doi:10.1017/S0305004100070158.
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