Caractéristiques d’Euler et groupes fondamentaux des variétés de dimension 4. (French) Zbl 0566.57011

Hausmann, Jean-Claude; Weinberger, Shmuel

Comment. Math. Helv. 60, 139-144 (1985).

Using a simple argument based on the multiplicative behaviour of the Euler characteristic with respect to a finite covering it is shown that if a 2-knot group has finite commutator subgroup T, then every abelian subgroup of T is cyclic (Theorem 5) and that there are finitely presentable superperfect groups which are not the fundamental groups of 4-dimensional homology spheres. (A result close to Theorem 5 was obtained by the reviewer [Bull. Aust. Math. Soc. 16, 449-462 (1977; Zbl 0352.57011)]. With regard to the closing problem of the article under review, it should be noted that C. M. Campbell and E. F. Robertson [Bull. Lond. Math. Soc. 12, 17-20 (1980; Zbl 0393.20020)] have shown that if p is a prime \(\geq 5\), then the finite group SL(2,p) has a presentation of deficiency 0; as it is superperfect it is the group of an homology 4-sphere.)

Reviewer: J.Hillman

MSC:

57Q45	Knots and links in high dimensions (PL-topology) (MSC2010)
57N13	Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
20F34	Fundamental groups and their automorphisms (group-theoretic aspects)

Keywords:

Euler characteristic; 2-knot group; commutator subgroup; finitely presentable superperfect groups; fundamental groups of 4-dimensional homology spheres

Citations:

Zbl 0418.20034; Zbl 0352.57011; Zbl 0393.20020