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
×
Y
and those of the spaces
X
and
Y
. 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.
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
×
Y
)
. (The argument applies equally to the simplicial or singular chain complexes.) We also have the tensor product complex
C
∗
(
X
)
⊗
C
∗
(
Y
)
, whose differential is, by definition,
δ
(
σ
⊗
τ
)
=
δ
X
σ
⊗
τ
+
(
−
1
)
p
σ
⊗
δ
Y
τ
for
σ
∈
C
p
(
X
)
and
δ
X
,
δ
Y
the differentials on
C
∗
(
X
)
,
C
∗
(
Y
)
.
Then the theorem says that we have chain maps
F
:
C
∗
(
X
×
Y
)
→
C
∗
(
X
)
⊗
C
∗
(
Y
)
,
G
:
C
∗
(
X
)
⊗
C
∗
(
Y
)
→
C
∗
(
X
×
Y
)
such that
F
G
is the identity and
G
F
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
×
Y
)
)
≅
H
∗
(
C
∗
(
X
)
⊗
C
∗
(
Y
)
)
.
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
H
∗
(
X
×
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.
