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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’.
##### MSC:
 57M20 Two-dimensional complexes (manifolds) (MSC2010) 57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc. 57M05 Fundamental group, presentations, free differential calculus
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##### References:
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