4-manifolds, group invariants, and \(l_2\)-Betti numbers
The present paper is a short survey concerning some interesting invariants for finitely presented groups, whose definitions and properties make use of 4-dimensional topology. In particular, the attention is fixed upon the Hausmann-Weinberger invariant \(p(G)\) (which is defined as the minimum of the Euler characteristic \(\chi(M)\), for all 4-manifolds \(M\) having the given group \(G\) as fundamental group: see [\textit{J.-C. Hausmann} and \textit{S. Weinberger}, Comment. Math. Helv. 60, 139-144 (1985; Zbl 0566.57011)]) and the \((\chi+\sigma)\)-invariant \(q(G)\) (whose definition involves both the Euler characteristic \(\chi(M)\) and the signature \(\sigma(M)\), \(M\) being a 4-manifold with fundamental group \(G\): see [\textit{D. Kotschick}, in: `Topology, geometry and field theory', 89-99 (1994; Zbl 0889.57029)]. Moreover, the paper resumes the cellular definition of \(l_2\)-cohomology and \(l_2\)-Betti numbers (as it appears, for example, in \textit{J. Cheeger} and \textit{M. Gromov}, Topology 25, 189-215 (1986; Zbl 0597.57020)]) and lists some good applications of \(l_2\)-methods to the previously considered group invariants. A detailed treatment of the subject may be found in [\textit{B. Eckmann}, Comment. Math. Helv. 69, No. 4, 501-511 (1994; Zbl 0857.57002)] and in [\textit{W. Lueck}, Topology 33, No. 2, 203-214 (1994; Zbl 0847.55004); Math. Ann. 309, No. 2, 247-285 (1997; Zbl 0886.57028)].
scientific article; zbMATH DE number 1140128