We now produce some cohomology classes of in degree , which will generate the whole cohomology.
For , let send to . Let be the pullback of the fundamental class along .
Note that , and , since the square of the fundamental class on vanishes.
The first claim is that these are dual to the .
Under the homology-cohomology pairing, if and , then
In particular, the set is linearly independent and forms a basis of , since we know this group is free of rank .
where the first map picks out and the second . The pairing above is given by the degree of this map.
If and , then this is the identity map, which has degree . Otherwise, taking gives a homotopy to the constant map.
We now know that as a ring, the cohomology of is generated by the . We introduce a graphical way to depict products of the . Such a product is represented by a graph with vertices . Each edge is oriented, and the set of edges is ordered. Multiple edges between vertices is allowed but loops are not.
To depict , we take the graph whose only edge is an edge from to . Products are then represented by unions. For example, is represented by the graph
The numbers on the edges denote the ordering of the edges.
If we swap the ordering of two adjacent edges or reverse an arrow, the class represented picks up a sign of . Moreover, since , any graph with a repeated edge represents .
There is one further relation, which morally is dual to the Jacobi identity.
We will prove this using the intersection product.
Recall that if is a compact manifold, then Poincaré duality tells us
Under this isomorphism, the cup product on induces a product on . If are represented by submanifolds that intersect transversely, then is represented by . This allows for a geometric way to compute the cup product structure of a compact manifold.
In the case of a non-compact manifold, there are two ways to fix Poincaré duality. One is to replace cohomology with compactly supported cohomology, but this is not what we want. Instead, we can replace homology with “locally finite homology”, also known as Borel–Moore homology, which is given by the homology of the complex of locally finite chains. In particular, arbitrary closed submanifolds of represent a Borel—More homology class.
Let be the submanifold of where are colinear and is in the middle. This is a submanifold of codimension , and represents a class in . We claim that up to a sign, it is (where ).
To compute the relevant class, we evaluate it on , and , which is computed by intersecting the relevant submanifolds. It does not intersect , because in , the point is at “infinity”, so cannot be in between and .
On the other hand, in and , there is exact one point where are colinear and is in the middle, and these points come with opposite signs.
Now and are disjoint, so the represented cohomology classes have zero product. So
We let be the abelian group generated by graphs on subject to the previous relations. This is a ring by taking unions.