What are null homologous cycles in an open set $Ω$ in $\mathbb C$? What is also an example that is not null-homologous?

  • $\begingroup$ Do you know what the winding number is? $\endgroup$ – rawbacon Nov 12 '19 at 18:41
  • $\begingroup$ no @Levi, i just need a concrete definition of a null homologous cycle and I should be fine. $\endgroup$ – KarinaMath Nov 12 '19 at 19:03
  • $\begingroup$ Thank you so much. A definition like this is exactly what I needed. @Levi $\endgroup$ – KarinaMath Nov 12 '19 at 19:09
  • $\begingroup$ @Levi Do you also have a definition of null homotopic curves in the same set? $\endgroup$ – KarinaMath Nov 12 '19 at 19:12
  • $\begingroup$ $\gamma$ is null-homotopic if there exists a continuous map (a "homotopy") $H: [0,1] \times [0,1] \rightarrow \Omega$ such that (i) $H(s, 0) = \gamma(0)$, (ii) $H(s, 1) = \gamma(1)$, (iii) $H(0, t) = \gamma(t)$, (iv) $H(1, t)$ is a constant path. (Null-homotopic means homotopic to a constant path). Intuitively, it means that you can deform $\gamma$ inside $\Omega$ such that in the end you are left with just a point. $\endgroup$ – rawbacon Nov 12 '19 at 19:16

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.

  1. A constant path is null-homologous in all $\Omega$.
  2. 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 $0$).
  3. 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.
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