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Andrews-Curtis operations and higher commutators of the relator group. (Andrews-Curtis-Operationen und höhere Kommutatoren der Relatorengruppe.) (German) Zbl 0742.57002
J. H. C. Whitehead proved that, given two \(n\)-dimensional (\(n>2\)) simple- homotopy equivalent polyhedra, there exists an \(n+1\)-dimensional polyhedron which collapses to both of the given ones. For \(n=2\) the question is essentially group-theoretic: it is known that 2-complexes which are simple-homotopy equivalent may be 3-deformed to certain standard complexes having presentations with an equal number of defining relators and the same relator subgroup \(N\). Indeed, corresponding relators \(R_ i\), \(S_ i\) differ by an element of the commutator subgroup \(N^{(1)}\) (i.e. \(R_ i\cdot S_ i^{-1}\in N^{(1)}\)). A result analogous to that of Whitehead for \(n=2\) boils down to deciding whether one set of relators may be carried to the other via a sequence of certain (Andrews-Curtis) operations. In this paper a step is taken along this route by showing that for two presentations as above, there exist operations which provide the generalized relationship between transformed relators \(R_ i\cdot S_ i^{-1}\in N^{(n)}\) for any \(n\) (where \(N^{(n)}\) is the nth derived group of \(N\)). The long-term goal is to understand the operations for various \(n\) in sufficient detail to ensure the eventual vanishing of the commutator ‘difference terms’.
57M20 Two-dimensional complexes (manifolds) (MSC2010)
57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
57M05 Fundamental group, presentations, free differential calculus
Full Text: DOI
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