On Fundamental Groups of Class VII Surfaces

The purpose of this note is to obtain a restriction on the fundamental groups of nonelliptic compact complex surfaces of class VII in Kodaira’s classification [9]. We recall that these are the compact complex surfaces with first Betti number one and no nonconstant meromorphic functions. This seems to be the class of compact complex surfaces whose structure is least understood. The first and simplest examples are the general Hopf surfaces [9], III. Then there are various classes of examples found by Inoue [5,6], and which have been studied in more detail in [11]. The only known topological restriction beyond the first Betti number is that intersection form in two-dimensional homology is negative definite. There seems to be little known as to how wide this class of surfaces is. We prove the following theorem.