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 $$X imes Y and those of the spaces $$X and $$Y. The theorem first appeared in a 1953 paper in the American Journal of Mathematics.

tatement of the theorem

The theorem can be formulated as follows. Suppose $$X and $$Y are topological spaces, Then we have the three chain complexes $$C_*(X), $$C_*(Y), and $$C_*(X imes Y) . (The argument applies equally to the simplicial or singular chain complexes.) We also have the "tensor product complex" $$C_*(X) otimes C_*(Y), whose differential is, by definition, :$$delta( sigma otimes au) = delta_X sigma otimes au + (-1)^p sigma otimes delta_Y au

for $$sigma in C_p(X) and $$delta_X, $$delta_Y the differentials on $$C_*(X),$$C_*(Y).

Then the theorem says that we have a chain maps

:$$F: C_*(X imes Y) ightarrow C_*(X) otimes C_*(Y), quad G: C_*(X) otimes C_*(Y) ightarrow C_*(X imes Y)

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

:$$H_*(C_*(X imes Y)) cong H_*(C_*(X) otimes C_*(Y)).

Consequences

The Eilenberg–Zilber theorem is a key ingredient in establishing the Künneth theorem, which expresses the homology groups $$H_*(X imes Y) in terms of $$H_*(X) and $$H_*(Y). 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; the answer is somewhat subtle.

References

*citation | last1=Eilenberg | first1=Samuel | last2=Zilber | first2=J. A. | title=On Products of Complexes | periodical=Amer. Jour. Math. | date=1953 | volume=75 | issue=1 | pages=200–204 | id=MathSciNet | id=52767 .
*citation | last=Hatcher | first=Allen | title=Algebraic Topology | date=2002 | publisher=Cambridge University Press | isbn=0-521-79540-0.