  Little is known about the global topology of the Fatou set $U(f)$ for
holomorphic endomorphisms $f: \mathbb{CP}^k \to \mathbb{CP}^k$, when $k >1$.
Classical theory describes $U(f)$ as the complement in $ \mathbb{CP}^k$ of the
support of a dynamically-defined closed positive $(1,1)$ current. Given any
closed positive $(1,1)$ current $S$ on $ \mathbb{CP}^k$, we give a definition
of linking number between closed loops in $\mathbb{CP}^k \setminus \supp S$ and
the current $S$. It has the property that if $lk(\gamma,S) \neq 0$, then
$\gamma$ represents a non-trivial homology element in $H_1(\mathbb{CP}^k
\setminus \supp S)$.
  As an application, we use these linking numbers to establish that many
classes of endomorphisms of $\mathbb{CP}^2$ have Fatou components with
infinitely generated first homology. For example, we prove that the Fatou set
has infinitely generated first homology for any polynomial endomorphism of
$\mathbb{CP}^2$ for which the restriction to the line at infinity is hyperbolic
and has disconnected Julia set. In addition we show that a polynomial skew
product of $\mathbb{CP}^2$ has Fatou set with infinitely generated first
homology if some vertical Julia set is disconnected. We then conclude with a
section of concrete examples and questions for further study.
