Gregory Arone: Tree complexes and obstructions to embeddings

Abstract:
Using the framework of the calculus of functors (a combination of manifold and orthogonal calculus) we define a sequence of obstructions for embedding a smooth manifold (or more generally a CW complex) M in \(R^d\). The first in the sequence is essentially Haefliger’s obstruction (or van Kampen obstruction in the case of CW complexes). The second one was studied by Brian Munson. We interpret the n-th obstruction as a cohomology of configurations of n points on M with coefficients in the homology of a complex of trees with n leaves. The latter can be identified with the cyclic Lie_n representation. We will illustrate the theory with some examples involving embedding 2-dimensional complexes in \(R^4\). The part having to do with CW complexes, especially embedding 2-complexes in \(R^4\), is joint with Slava Krushkal. All of this is very much work in progress.