Null-homologous cycles in an open set What are null homologous cycles in an open set $Ω$ in $\mathbb C$? What is also an example that is not null-homologous?
 A: Let $\Omega \subset \mathbb C$. A closed curve $\gamma: [0,1] \rightarrow \Omega$ is null-homologous if the winding number of $\gamma$ around any point in $\mathbb C \setminus \Omega$ is zero. The winding number of $\gamma$ around $z_0$ is the number of times $\gamma$ wraps around zero. Using complex analysis, we can define the winding number $n(\gamma, z_0)$ of $\gamma$ around $z_0$ to be
$$n(\gamma,z_0)=\frac{1}{2\pi i}\int_{\gamma}\frac{1}{z}\,dz.$$
It is not obvious that these two notions coincide (this has to do with homotopy-invariance of path integrals), but they sure do.
Here are some instructive examples.


*

*A constant path is null-homologous in all $\Omega$.

*The standard embedding of the circle $t \mapsto e^{2\pi i t}$ is null-homologous in $\mathbb C$ (trivially because there is no point in $\mathbb C \setminus \mathbb C$), but not null-homologous in $\mathbb C \setminus \{0\}$ (because it winds once around $0$).

*If you have seen path-homotopy then you might think that null-homologous and null-homotopic (being "continuously shrinkable to one point") are the same. It turns out that being null-homologous is actually weaker. Indeed, one can hang up a picture using two nails and a string such that removing any one of the nails causes the picture to fall down. Let $\Omega = \mathbb C \setminus \{-1, 1\}$. Let $\gamma$ be the following curve: trace an $\infty$-shape around these two points: once around $1$ counterclockwise, around $-1$ counterclockwise, then again around $1$ but clockwise, and then finally around $-1$ clockwise. Then $\gamma$ is null-homologous in $\Omega$, but not null-homotopic.

