Next Previous Contents

## 8. Products

Let all modules be R-modules for some principal ideal domain R and let tensor products be over R.

## 8.1 The product of two chain complexes

Given a product operation like tensor (or *, see below), we can form the product of two chain complexes A and B, say A tensor B, by taking all products A_i tensor B_j and letting (A tensor B)_k = directsum A_i tensor B_j where the sum is over the i,j with i+j=k. We also need a differential, and define d(a tensor b) = da tensor b + (-1)^i a tensor db when a lies in A_i. (Indeed, dd = 0.)

There is a natural map H(A) tensor H(B) -> H(A tensor B) sending a tensor b to a tensor b. (Indeed, if a is a boundary, say a = dc, and b is a cycle, then a tensor b = d(c tensor b) is a boundary, so this map is well-defined.)

## 8.2 The Eilenberg-Zilber theorem

For a topological space X, let us write S(X) for the singular chain complex of X.

Theorem Let X and Y be topological spaces. Then S(X × Y) is chain equivalent to S(X) tensor S(Y).

## 8.3 Künneth formula

Let C,D be chain complexes, where C*D is acyclic. Then there is a short split exact sequence 0 -> H(C) tensor H(D) -> H(C tensor D) -> (H(C) * H(D)) -> 0 with maps of degree 0, 0, -1, respectively. This expresses H(C tensor D) in terms of H(C) and H(D).

## 8.4 Homology with coefficients

Instead of doing homology with integral coefficients, we can use coefficients from an Abelian group G. The resulting chain complexes is C tensor G. The (co)homology obtained in this way is denoted H(C;G).

Let C be a free chain complex (i.e., each C_i is free). Then there is a short split exact sequence 0 -> H(C) tensor G -> H(C tensor G) -> H(C) * G -> 0 with maps of degree 0, 0, -1, respectively. This expresses H(C;G) = H(C tensor G) in terms of H(C).

## 8.5 The cross product

There is a homomorphism H(A;G) tensor H(B;H) -> H(A tensor B; G tensor H) defined in the obvious way (using the identification (A tensor G) tensor (B tensor H) = (A tensor B) tensor (G tensor H)). This gives rise to a multiplication on homology called the cross product.

Similarly, one has a cross product on cohomology. In particular, following Eilenberg-Zilber, there is for a topological space X a map H*(X;G) tensor H*(X;H) -> H*(X;G tensor H). But the diagonal map x -> (x,x) from X into X × X gives rise to a homomorphism H*(X × X; K) -> H*(X; K), and composing that with the cross product we get the cup product H*(X;G) tensor H*(X;H) -> H*(X; G tensor H), and, in particular, H*(X) tensor H*(X) -> H*(X).

The particular incarnation of the cup product that we described above is obtained by taking the map t : S(X) -> S(X) tensor S(X) defined by t(u) = sum head(u,i) tensor tail(u,j) where the sum is over the i,j with i+j=k and head(u_0...u_k, i) = u_0...u_i and tail(u_0...u_k, j) = u_(k-j)...u_k.

Next Previous Contents