Cockcroft-Swan theorem for projective crossed chain complexes
\textit{W. H. Cockcroft} and \textit{R. G. Swan} [Proc. Lond. Math. Soc. (3) 11, 193--202 (1961; Zbl 0096.37603)] proved that if \(f: C \to C'\) is a chain map between two \(n\)-dimensional projective (free) chain complexes that induces isomorphisms \(f_* : H_j(C)\cong H_j (C') \) for all \(j<n\), then by adding elementary projective (free) complexes to \(C\) and \(C'\), the chain map \(f\) can be extended to a chain isomorphism. An elementary complex is a complex \(D\) with \(D_i=0\;(i\neq q, q+1)\) for some \(q\), and \(D_{q+1}=D_q\) with boundary operator \(\partial _{q+1}=\mathrm{id}\). This theorem was proved in [loc. cit.] and the process is called the stabilization of \(f\). In [Functions on manifolds. Algebraic and topological aspects. Translated from the Russian by V. V. Minachin. Translation edited by Simeon Ivanov. Providence, RI: American Mathematical Society (1993; Zbl 0791.57001)], \textit{V. V. Sharko} proved an analog of Cockcroft-Swan theorem for free crossed chain complexes. In the paper under review, an analog of the Cockcroft-Swan theorem is proved for \(n\)-dimensional projective crossed chain complexes. In this setting, the stabilization of a projective crossed chain complex \((P_i, G, \partial _i)\) is done by replacing the group \(G\) with the group \(G\ast F\) for some finitely generated free group \(F\) and by adding elementary projective complexes at higher dimensions.
scientific article; zbMATH DE number 6632552
scientific article