Isaac Mabillard (Institute of Science and Technology Austria)

2015/01/16, 16:00

TU Berlin, at room MA313 (Straße des 17. Juni 136)

The Kuratowski theorem provides a nice criterion for graph planarity, ie, to decide whether a simplicial $1$-complex can be embedded into $\mathbb{R}^2$.

A natural generalization of the problem is to find a criterion to decide whether a simplicial $n$-complex $K$ can be embedded into $\mathbb{R}^{2n}$. This is what the van Kampen obstruction cocycle gives us. By using standard tricks in PL topology, one can show that $K$ is embeddable if and only if (the class of) its cocycle is zero.

This is (maybe?) surprising because embeddability is a geometric question, whereas a cocycle is an algebraic object, but it still carries enough information to solve the geometric problem.