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.