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.