On T. Petrie's problem concerning homology planes (Q1174388)

``Homology plane'' here means a smooth affine complex algebraic surface with the homology of a point. A suggested method of constructing examples is to take irreducible curves \(F\) and \(G\) in \(\mathbb{P}^ 2(\mathbb{C})\), each homeomorphic to \(\mathbb{P}^ 1(\mathbb{C})\), having just 2 intersection points which are smooth on each of \(F\) and \(G\), and mutually coprime degrees \(m\leq n\), and form \(X\) by deleting \(F\) and \(G\) from \(\mathbb{P}^ 2\). We assume \(X\) not isomorphic to \(\mathbb{C}^ 2\). The main part of the paper is occupied by a combinatorial argument (involving many case distinctions) in which the Miyaoka inequality on a suitable blow-up is used to obtain an inequality between combinatorial invariants, and this is in turn used to show that in the above situation, \(m\leq 3\). These cases are then discussed further (e.g. \(n\) also is bounded if \(m\geq 2)\), but inconclusively: it is unclear whether or not examples exist with \(m=3\), and the only three examples given with \(m=2\) (all with \(n=3)\) where previously known. The appendix considers the case \(m=1\) above and obtains two doubly- infinite families of examples, given by explicit equations. A singly- infinite subfamily is shown to consist of simply-connected (hence contractible) surfaces.