Little is known about the global topology of the Fatou set U(f) for holomorphic endomorphisms $f: ℂℙ^k → ℂℙ^k$, when k >1. Classical theory describes U(f) as the complement in $ℂℙ^k$ of the support of a dynamically defined closed positive (1,1) current. Given any closed positive (1,1) current S on $ℂℙ^k$, we give a definition of linking number between closed loops in $ℂℙ^k ∖ supp S$ and the current S. It has the property that if lk(γ,S) ≠ 0, then γ represents a non-trivial homology element in $H₁(ℂℙ^k ∖ supp S)$.
As an application, we use these linking numbers to establish that many classes of endomorphisms of ℂℙ² 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 ℂℙ² 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 ℂℙ² 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.