Afshin Goodarzi: On embedding of 2-dimensional simplicial complexes

Every graph can be drawn in the Euclidean 3-space without self intersection. A planar graph is a graph that can be drawn in the plane without self intersection. The concept of planarity has been of interest to mathematicians ever since the subject of graph theory was founded. For instance, the impossibility of having more than 3n-6 edges for a planar graph on n vertices was mentioned in a letter from Euler to Goldbach in 1750. A topological characterisation of planarity was given by Kuratowski. Since then other characterisations of planarity have been given. Among them one can mention the more combinatorial approaches by Mac Lane, and the more topological approach of Hanani and Tutte.

What about higher dimensional objects? Let X be a 2-dimensional complex with n vertices, e edges, and t triangles. If X can be embedded in the Euclidean 3-space, then t is at most 2(e-n). 

This talk will be concerned with 2-dimensional simplicial complexes. Every 2-dimensional simplicial complex admits an embedding in the 5-dimensional Euclidean space. This result is tight; there are 2-dimensional simplicial complexes that do not embed in the 4-dimensional Euclidean space. I will discuss analogues of some characterisation of planarity for 2-dimensional complexes. This leads to  obstructions to embeddability of 2-dimensional complexes in all possible dimensions (2,3, and 4). 

The new results that will be presented are from a recent joint work with Anders Björner.