1. The cross product

products :

SHi(X) × SHj (Y ) → SHi+j (X × Y )

and

SHi(X; Z/2) × SHj (X; Z/2) → SHi+j (X × Y ; Z/2) which are defined as

[S, g] × [S , g ] := [S × S , g × g ].

The following Proposition follows from the definition of the ×-product:

Proposition 10.1. The ×-products are bilinear and associative.

Since the ×-products are bilinear they induce maps from the tensor product

SHi(X; Z/2) Z/2 SHj (Y ; Z/2) −→ SHi+j (X × Y ; Z/2)

and

SHi(X) SHj (Y ) −→ SHi+j (X × Y ).

(We denote the tensor product of abelian groups by and of F -vector spaces by F .)

We sum the left side over all i, j with i+j = k to obtain homomorphisms

× : SHi(X) SHj (Y ) → SHk(X × Y )

i+j=k

and

× : SHi(X; Z/2) Z/2 SHj (Y ; Z/2) → SHk(X × Y ; Z/2).

i+j=k

It would be nice if these maps were isomorphisms. For Z/2-homology, we will show this under some assumptions on X, but for integral homology these assumptions are not su cient. The idea is to fix Y and to consider the functor

SHkY (X) := SHk(X × Y ) where for f : X → X we define

f : HkY (X) → HkY (X )

by (f × id) . This is obviously a homology theory: the Mayer-Vietoris sequence holds since

(U1 × Y ) (U2 × Y ) = (U1 U2) × Y,

(U1 × Y ) ∩ (U2 × Y ) = (U1 ∩ U2) × Y.

Furthermore this is a homology theory with compact supports.

For X a point the maps × above are isomorphisms. Thus we could try to prove that they are always an isomorphism for nice spaces X by applying

the comparison result Corollary 9.4 if

X −→ SHi(X) SHj (Y ) =: hYk (X)

i+j=k

were also a homology theory and, similarly, if

X −→ SHi(X; Z/2) Z/2 SHj (Y ; Z/2) =: hYk (X; Z/2)

i+j=k

were a homology theory. Here, for f : X → X , we define

f = ((f id) : SHi(X) SHj (Y ) → SHi(X ) SHj (Y )).

i+j=k

The homotopy axiom is clear but the Mayer-Vietoris sequence is a problem. It would follow if for an exact sequence of abelian groups

f g

A → B → C

and an abelian group D the sequence

were exact. But this is in general not the case. For example consider

·2

0 → Z −→ Z

and D = Z/2 giving

·2

0 → Z/2Z −→ Z/2Z

which is not exact since ·2 : Z/2Z → Z/2Z is 0. If instead of abelian groups we work with vector spaces over a field F , the sequence

is exact. It is enough to show this for short exact sequences 0 → A → B → C → 0 by passing to the image in C and dividing out the kernel in A. Then

there is a splitting s : C → B with gs = id and a splitting p : B → A with pf = id. These splittings induce splittings of

The sequence is also exact if D is a torsion-free finitely generated abelian

group. Namely then D Zr for some r. It is enough to check exactness

=

for r = 1, where it is trivial since A Z A. For larger r we use that

=

A (D D ) (A D) (A D ).

=

Thus, if all homology groups of Y are finitely generated and torsion-free, the functor hYk (X) is a generalized homology theory. And since SHk(X; Z/2) is a Z/2-vector space we conclude that for any fixed space Y the functor hYk (X; Z/2) is a homology theory. To obtain some partial information about the integral homology groups of a product of two spaces, if SHk(Y ) is not finitely generated and torsion-free, we define rational homology groups.

Definition: SHm(X; Q) := SHm(X) Q. For f : X → Y we define f : SHm(X; Q) → SHm(Y ; Q) by f id : SHm(X) Q → SHm(Y ) Q.

By the considerations above the rational homology groups define a homology theory called rational homology. Since the rational homology groups are Q-vector spaces (scalar multiplication with λ Q is given by λ(x μ) := x λμ), the functor

X −→ SHi(X; Q) SHj (Y ; Q) =: hYk (X; Q)

i+j=k

is a homology theory. By construction it has compact supports.

2. The K¨unneth theorem

To apply Corollary 9.4 we have to check that the maps × : hYk (X; Z/2) →

SHkY (X; Z/2), × : hYk (X) → SHkY (X) and × : hYk (X; Q) → SHkY (X; Q) commute with induced maps and the boundary operator in the Mayer-

Vietoris sequence, in other words, that these maps are natural transformations. The proof is the same in both cases and so we only give it for Z/2-homology:

Lemma 10.2. The maps

× : hYk (X; Z/2) → SHkY (X; Z/2) define a natural transformation.

Proof: Everything is clear except the commutativity in the MayerVietoris sequences. Let U1 and U2 be open subsets of X and consider for i + j = k the element [S, f] [Z, g] SHi(U1 U2) SHj(Y ). By our definition of the boundary operator in the Mayer-Vietoris sequence of SHi(X) we can decompose the stratifold S (after perhaps changing it by a bordism) as S = S1 S2 with ∂S1 = ∂S2 =: Q, where f(S1) U1 and f(S2) U2. Then d([S, f]) = [Q, f|Q]. Thus d([S, f] [Z, g]) = [Q, f|Q] [Z, g]. On the other hand [S, f] × [Z, g] = [S × Z, f × g] and, since S × Z = (S1 S2) × Z, we conclude: d([S × Z, f × g]) = [Q × Z, f|T × g]. Thus the diagram

commutes.

q.e.d.

Now the K¨unneth Theorem is an immediate consequence of Corollary

9.4:

Theorem 10.3. (K¨unneth Theorem) Let X be a nice space (see page

95). Then for F = Q or Z/2Z

× : SHi(X; F ) F SHj (Y ; F ) → SHk(X × Y ; F )

i+j=k

is an isomorphism. The same holds for integral homology if for all j the groups SHj (Y ) are torsion-free and finitely generated.

We note that the Kunneth theorem stated here holds for all spaces X which admit decompositions as finite CW -complexes since all these spaces are nice. In a later chapter we will identify the stratifold homology of CW - complexes with the homology groups defined in a traditional way using simplices. The world of chain complexes is more appropriate for dealing with the K¨unneth Theorem and there one obtains a general result computing the integral homology groups of a product of CW -complexes.