A note on fake surfaces and universal covers
For a finitely presented group \(G\), the second cohomology \(H^2(G;\mathbb{Z}G)\) is known to be torsion-free, and has long been conjectured to be free abelian. Fundamental groups of \(3\)-manifolds satisfy the conjecture. In this paper the author extends previous work to provide a sufficient condition for certain groups to satisfy the conjecture. Every finitely presented group \(G\) can be realized as the fundamental group of a fake surface \(X\), that is, a \(2\)-complex in which the star of each vertex is the cone either on a circle, on the suspension of three points, or on the \(1\)-skeleton of a tetrahedron. The latter are called Type~III vertices. The main result concerns surfaces \(X\) in which each component of the set of non-manifold points contains at most one Type~III vertex. A component containing one Type~III vertex must be a figure-\(8\) with fundamental group generated by two imbedded loops \(\gamma_1\) and \(\gamma_2\), and these loops generate a subgroup of \(\pi_1(X)\). The main result states that if for each such component, the resulting subgroup is isomorphic either to the free product \(\langle \gamma_1\rangle * \langle \gamma_2\rangle\) or to the free abelian group \(\langle \gamma_1\rangle \oplus \langle \gamma_2\rangle\), and another geometric condition is satisfied whenever \(\langle \gamma_i\rangle\) has finite odd order, then there exists a compact polyhedron \(K\) with \(X\subseteq K\) and \(\pi_1(X)\cong \pi_1(K)\), such that \(K\) can be thickened to a \(3\)-manifold. Consequently, such \(\pi_1(X)\) satisfy the conjecture.
scientific article; zbMATH DE number 1879141