Identities between sets of relations
Given a presentation \({\mathcal R}=\langle X| R\rangle\) of a group \(G\) we may construct the free crossed module \(\rho: C(R)\to F\) associated with the inclusion of \(R\) into \(F\), the free group on \(X\). By a theorem of \textit{J. H. C. Whitehead} we may interpret \(\rho\) as a boundary homomorphism of homotopy groups \(\pi_ 2(K\langle X| R\rangle,K^ 1)\to\pi_ 1(K^ 1)\). Then \(\ker\rho=\pi_ 2({\mathcal R})\) is a \(G\)- module, the module of identities under the relations \(R\). The author studies \(\pi_ 2({\mathcal P})\), where \({\mathcal P}= \langle X| R\cup S\rangle\), with \(R\cap S=\emptyset\). His section 3 exploits the decomposition of \(\pi_ 2(K\langle X| R\cup S\rangle,K^ 1)\) as a Peiffer product of \(\pi_ 2(K\langle X| R\rangle,K^ 1)\) and \(\pi_ 2(K \langle X| S\rangle,K^ 1)\) to examine identities between the sets of relations measured by the \(F\)-module \(\langle\langle R\rangle\rangle \cap \langle\langle S\rangle\rangle/[ \langle\langle R \rangle\rangle, \langle\langle S\rangle\rangle]\), where the double brackets denote normal closure in \(F\). Section 4 examines the action of \(\langle\langle S\rangle\rangle\) on the module of identities among the relations \(R\), considering \(S\)-balanced and \(S\)-coupled identities and interpretations of third homology groups in terms of such identities. The generalized Van Kampen theorem [\textit{R. Brown} and \textit{J.-L. Loday}, Topology 26, 311-335 (1987; Zbl 0622.55009)] implies that the triad homotopy group \(\pi_ 3(K\langle X| R\cup S\rangle;K\langle X| R\rangle,K\langle X| S\rangle)\) is a non-abelian tensor product of the relative groups \(\pi_ 2(K\langle X| R\rangle,K^ 1)\) and \(\pi_ 2(K\langle X| S\rangle,K^ 1)\). The tensor product permits a unified treatment of \(S\)-balanced identities among the relations \(R\) and of \(R\)-balanced identities among the relations \(S\). In section 5 computations in the form of exact sequences of \(F/[\langle\langle R\rangle\rangle, \langle\langle S\rangle\rangle]\)-modules are given for the consequent measure of balanced identities between sets of relations.
boundary homomorphism of homotopy groups
module of identities under the relations
scientific article; zbMATH DE number 166175
scientific article