4 The homology–cohomology pairing
To compute the homology–cohomology pairing, since is generated by the , it suffices to compute the composites
for forests .
If and are in different trees, by taking the limit , we see that this map is homotopic to the constant map , hence is nullhomotopic. Otherwise, if they meet at , then it is easy to see that up to a sign, this is projection onto the th factor.
Let be an -forest and an -graph. Then under the homology–cohomology pairing, is iff
For every edge in , the corresponding leaves in are in the same component
The map sending an edge to the meet of leaves gives a bijection between edges of and vertices of .
Otherwise, it is zero.
The lemma is then clear from the previous identification.
Using this explicit description of the pairing, let us show that the maps and are isomorphisms. We already know that the second map is surjective. Our plan is as follows:
Write down a spanning set of and .
Show that the homology–cohomology pairing pairs these elements perfectly.
Deduce the maps must be injections and , so they are both isomorphisms.
The following lemma follows from applying the Jacobi and Arnold identities:
is spanned by “tall” forests, i.e. forests whose trees look like
where the leftmost vertex has minimal label amongst the leaves. is spanned by “long graphs” whose components look like
Each of these elements are specified by partitions of together with some ordering data.
The pairing of a tall forest and a long graph is if they correspond to the same partition, and otherwise.
and are isomorphisms.