What is … a van Kampen obstruction cocycle?

This page hosts information on Isaac Mabillard's talk "What is … a van Kampen obstruction cocycle?" at the "What is …?" seminar. The talk will take place on Friday, January 16, 4:00pm in room MA 313 at TU Berlin.

Abstract

The Kuratowski theorem provides a nice criterion for graph planarity, ie, to decide whether a simplicial 1-complex can be embedded into 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 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.

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