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. One possible route to a proof is the acyclic model theorem.
Statement of the theorem
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
An important generalisation to the non-abelian case using crossed complexes is given in the paper by 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 Brown and Higgins on classifying spaces.
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.
- Eilenberg, Samuel; Zilber, J. A. (1953), "On Products of Complexes", Amer. Jour. Math., American Journal of Mathematics, Vol. 75, No. 1, 75 (1), pp. 200–204, doi:10.2307/2372629, JSTOR 2372629, MR 52767.
- Hatcher, Allen (2002), Algebraic Topology, Cambridge University Press, ISBN 0-521-79540-0.
- Tonks, Andrew (2003), "On the Eilenberg-Zilber theorem for crossed complexes", Jour. Pure Applied Algebra, 179, pp. 199–230.
- Brown, Ronald; Higgins, Philip J. (1991), "The classifying space of a crossed complex", Proc. Camb. Phil. Soc., 110, pp. 95–120, doi:10.1017/S0305004100070158.