The surjectivity problem for one-generator, one-relator extensions of torsion-free groups. (English) Zbl 1014.20015
If a group \(\widehat G\) is obtained from a group \(G\) by adding \(n\) generators and \(n\) relators then the surjectivity problem is whether the natural homomorphism \(G\to\widehat G\) is surjective. This was raised by M. M. Cohen [Topology 16, 79-88 (1977; Zbl 0351.57002)] and independently by W. Metzler [J. Reine Angew. Math. 285, 7-23 (1976; Zbl 0325.57003)]. Here it is proved that the natural map \(G\to\widehat G\), where \(G\) is a torsion-free group and \(\widehat G\) is obtained by adding a new generator \(t\) and a new relator \(w\), is surjective only if \(w\) is conjugate to \(gt\) or \(gt^{-1}\) where \(g\in G\). One of the corollaries is that for a CW-complex \(\widehat L\), that is obtained from a CW-complex \(L\) by attaching first a \(1\)-cell and then a \(2\)-cell, the inclusion map \(j\colon L\to\widehat L\) is a simple homotopy equivalence if \(j\) induces a surjection \(j_*\colon\pi_1L\to\pi_1\widehat L\). The proof uses methods and results of A. A. Klyachko [Commun. Algebra 21, No. 7, 2555-2575 (1993; Zbl 0788.20017)].
Reviewer: Wolfgang Heil (Tallahassee)
MSC:
| 20F05 | Generators, relations, and presentations of groups |
| 20E22 | Extensions, wreath products, and other compositions of groups |
| 57M20 | Two-dimensional complexes (manifolds) (MSC2010) |
| 57Q10 | Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc. |
Keywords:
surjectivity problem; torsion-free groups; Whitehead torsion; Kervaire conjecture; simple homotopy equivalences; CW-complexesReferences:
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