Hybrid talk. For zoom details contact Barthel, Ozornova, Ray, Teichner.

The theory of non-planar graphs has a higher dimensional generalization---essentially due to vanKampen---which describes whether a given n-complex embeds in 2n-dimensional space. It gives a complete description for n>2. In the exceptional case n=2, Freedmann, Krushkal and Teichner constructed examples showing there are additional embedding obstructions. In this talk, I will describe this classical (and semi-classical) embedding theory and a new class of 2-complexes that do not embed in R^4. A novel feature is that they do embed in some (mod 2)-homology 4-spheres, and that each one admits a sequence of immersions f_k "hiding'' the obstruction to embedding from being detected via k-step nilpotent groups. This is joint work with Tam Nguyen Phan.

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