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Tree complexes and obstructions to embeddings.

Presented by: 
Gregory Arone
Tuesday 31st July 2018 - 15:30 to 16:30
INI Seminar Room 2
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. 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. When M is a union of circles, we conjecture that our obstructions are closely related to Milnor invariants. When M is of dimension 2 and d=4, we speculate that our obstructions are related to ones constructed by Schneidermann and Teichner. This is very much work in progress.

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons