Why are topological properties described by surface terms?

Question

An example are the anomalies in abelian and non-abelian gauge quantum field theories. 

For example, the abelian anomaly is $\tilde {F}_{\mu\nu}F^{\mu\nu}$ and the integral over this quantity is a topological invariant which measures a topological characteristic of the gauge field $A_\mu$.

All such quantities can be rewritten as total derivatives and then, using Gauss' law transformed into a surface integral. 

**What's the intuitive reason that quantities which describe topological properties can always be written as surface integrals?**

 Formulated a bit differently: Why are topological properties always completely encoded in the boundary of the system? 

Comments

Locally (i.e., in the bulk), topology cannot enter as all manifolds are locally diffeomorphic to the unit ball. 

Answer 1

Effectively this is just Stokes' theorem: The integral over spacetime of an exact differential form is the integral of its potential over the boundary. The only subtlety is that the interesting topological invariants are only locally exact, not globally. But the analogous statement remains true with differential forms enhanced to higher gerbes with connection. Instead of Chern-Simons forms integrated over the boundary one then gets the volume holonomy of higher Chern-Simons gerbes over the boundary. That the analogue of Stokes's theorem still applies in this relevant generality is proposition 4.4.130 (p. 532) in arxiv:1310.7930 .