Afshin Goodarzi: Obstructions to Codimension One Embedding of Simplicial Complexes

Abstract

It is known since before 1930 that a d-dimensional simplicial complex is embeddable into the Euclidean (2d+1)-space. In his 1932 article, van Kampen showed that this result is the best possible, by presenting d-dimensional complexes that do not embed into the Euclidean (2d)-space.

The major question concerning embeddings of the complexes is: "Given a d-dimensional simplicial complex X and an integer k between d and 2d. When does X embed in the Euclidean k-space?".

The cases when k=2d or k=d+1 are probably the most intensively investigated cases:

- When k=2d (and d is different from 2), the problem is decidable. Actually, there exists an algorithm based on van Kampen's ideas that solve this problem in a polynomial time. 
- When k=d+1 and d>3, the problem is not even algorithmically decidable. This was shown by Matousek, Tancer and Wagner in 2009 using a significant result of Novikov on sphere recognition.

In this talk, which is based on a recent joint work with Anders Björner, we give a homological obstruction to codimension one embedding, i.e. k=d+1 case. Some combinatorial corollaries in low dimensions will be presented. Furthermore, we provide a new upper bound for the number of facets of d-dimensional complex that embeds into Euclidean (d+1)-space.