PDF《Cohomology is an algebraic variant》

Cohomology is an algebraic variant of homology, the result of a simple dualiza- tion in the de?nition.

Not surprisingly, the cohomology groups Hi (X) satisfy axioms much like the axioms for homology, except that induced homomorphisms go in the opposite direction as a result of the dualization. The basic distinction between homol- ogy and cohomology is thus that cohomology groups are contravariant functors while homology groups are covariant. In terms of intrinsic information, however, there is not a big di?erence between homology groups and cohomology groups. The homol- ogy groups of a space determine its cohomology groups, and the converse holds at least when the homology groups are ?nitely generated. What is a little surprising is that contravariance leads to extra structure in co- homology. This ?rst appears in a natural product, called cup product, which makes the cohomology groups of a space into a ring. This is an extremely useful piece of additional structure, and much of this chapter is devoted to studying cup products, which are considerably more subtle than the additive structure of cohomology. How does contravariance lead to a product in cohomology that is not present in homology? Actually there is a natural product in homology, but it takes the somewhat di?erent form of a map Hi(X)*Hj(Y ) →Hi+j(X*Y ) called the cross product. If both X and Y are CW complexes, this cross product in homology is induced from a map of cellular chains sending a pair (ei , ej ) consisting of a cell of X and a cell of Y to the product cell ei *ej in X*Y . The details of the construction are described in §3.B. Taking X = Y , we thus have the ?rst half of a hypothetical product Hi(X)*Hj(X) →Hi+j(X*X) →Hi+j(X) The di?culty is in de?ning the second map. The natural thing would be for this to be induced by a map X*X→X . The multiplication map in a topological group, or more generally an HCspace, is such a map, and the resulting Pontryagin product can be quite useful when studying these spaces, as we show in §3.C. But for general X , the only

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3 Cohomology natural maps X*X→X are the projections onto one of the factors, and since these projections collapse the other factor to a point, the resulting product in homology is rather trivial. With cohomology, however, the situation is better. One still has a cross product Hi (X)*Hj (Y ) →Hi+j (X*Y ) constructed in much the same way as in homology, so one can again take X = Y and get the ?rst half of a product Hi (X)*Hj (X) →Hi+j (X*X) →Hi+j (X) But now by contravariance the second map would be induced by a map X→X*X , and there is an obvious candidate for this map, the diagonal map ?(x) = (x, x). This turns out to work very nicely, giving a well-behaved product in cohomology, the cup product. Another sort of extra structure in cohomology whose existence is traceable to contravariance is provided by cohomology operations. These make the cohomology groups of a space into a module over a certain rather complicated ring. Cohomology operations lie at a depth somewhat greater than the cup product structure, so we defer their study to §4.L. The extra layer of algebra in cohomology arising from the dualization in its def- inition may seem at ?rst to be separating it further from topology, but there are many topological situations where cohomology arises quite naturally. One of these is Poincar? e duality, the topic of the third section of this chapter. Another is obstruction theory, covered in §4.3. Characteristic classes in vector bundle theory (see [Milnor &

Stashe? 1974] or [VBKT]) provide a further instance. From the viewpoint of homotopy theory, cohomology is in some ways more basic than homology. As we shall see in §4.3, cohomology has a description in terms of homotopy classes of maps that is very similar to, and in a certain sense dual to, the de?nition of homotopy groups. There is an analog of this for homology, described in §4.F, but the construction is more complicated. The Idea of Cohomology Let us look at a few low-dimensional examples to get an idea of how one might be led naturally to consider cohomology groups, and to see what properties of a space they might be measuring. For the sake of simplicity we consider simplicial cohomology of ? complexes, rather than singular cohomology of more general spaces. Taking the simplest case ?rst, let X be a