Topological invariants by integrals

Some topological invariants that can be found e.g. in knot theory can be represented as integrals (Example: Integral for computing the Gauss linking number). Another example is the complex plane with poles: If the poles are defined by Terms like $\frac{1}{z-z_i}$ with some natural number $i$ that is indexing the pole then the number of poles (topological invariant) can be expressed as:

$N = \int_C \frac{f(z)dz}{2 \pi i}$.

In differential geometry the total curvature of a curve $c(t)$ parametrized by $t \in [0,1]$ is obtained by

$TC[c(t)]= \int_0^1 \kappa(t) dt$

with the curvature of the curve $\kappa(t)$. Also the Gauss-Bonnet Theorem computes the Euler characteristic from an integral.

Question: Can be expressed other topological invariants in Terms of integrals over (differential-)geometrical functions like curvature, Gauss curvature, etc.? What Basic ideas are used to link topological invariants with such integrals (how one can derive such identities)?

• I think generally, the opposite question is asked. Namely, it is usually more natural to ask whether or not a differential-geometric construction is related to a topological invariant rather than the other way around, since differential structures come to us with a topology already on them. Apr 3 '15 at 22:45