We produce some homology classes associated to trees.
Let . An -tree is a binary tree with a root vertex and leaves labelled by elements of (each label is used exactly once). We also distinguish between the left and right branches of the binary tree.
By convention, the term “vertex” does not refer to the leaves. Thus, the number of vertices is the number of leaves . We write for the number of vertices of .
The following is are trees with 1 and 3 vertices respectively:
Fix . Given an -tree , define a map as follows. If , put at some fixed point “at infinity”. For the rest, we define this by example.
, we send to the configuration
We abbreviate this as .
we send to
The fundamental class of gives us a corresponding homology class in , which we still call .
The classes in satisfy the relations
for any (sub)trees
To simplify notation, we only prove the second identity in the case where are trivial and , i.e. we show that
The general case admits the exact same proof.
Consider the submanifold of consisting of points satisfying:
The perimeter of the triangle spanned by is .
The sides of the triangle have length at least .
One observes that this manifold has three boundary components, achieved when one of the sides have length exactly . One then sees that this component is homotopic to for some in the identity. For example, the component where the – side has length is homotopic to the first term.
An -forest is a collection of -trees where each element in is used exactly once.
If is an -tree, we define in the same way as , where the position of is specified by the tree that contains , and the th tree is translated by .
Let be the abelian group generated by -forest subject to the (anti)commutativity and the Jacobi identity.
There is then a map , which we will show to be an isomorphism.