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Generalized group presentation and formal deformations of CW complexes. (English) Zbl 0764.57012
If \(f: K^ n\to L^ n\) is a simple-homotopy equivalence of connected, finite CW-complexes, then by C. T. C. Wall [Proc. Lond. Math. Soc., III. Ser. 16, 342-354 (1966; Zbl 0151.313)] there is a formal deformation \(K^ n\overset{n+1}\curvearrowright L^ n\), provided \(n\geq 3\). For \(n=2\), the concepts of equivalence by 3-deformations and simple-homotopy might disagree. (Andrews-Curtis problem). The present paper presents an (algebraic) treatment of the whole picture which combines Whitehead’s approach of homotopy systems with R. Peiffer’s \(\Sigma\)-systems; the latter associate to a 3-complex, generators, relators and identities [Math. Ann. 121, 67-99 (1949; Zbl 0040.150)]. A geometric result of the author’s work is that, if a 3-cell \(e^ 3\) has several free faces, then there is a characteristic map for \(e^ 3\) which realizes the freeness of all these faces simultaneously. The analogous question for faces of \(n\)-cells seems to be open.
Initially, the reviewer stumbled over the use of topological embeddings \(D^{n-1}\hookrightarrow\partial D^ n\) to define free faces, as (for \(n\geq 4)\) the result might be a solid wild ball in \(\partial D^ n\), and one has to worry whether there exists a deformation retraction for a collapse. But he confirmed himself that this can nevertheless be proved (use a subdisc of \(D^{n-1}\) in standard position and apply M. Brown’s Schoenflies theorem). If this point wasn’t just overlooked by the author, a corresponding remark would have been appropriate.

57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
57M05 Fundamental group, presentations, free differential calculus
20F05 Generators, relations, and presentations of groups
57Q05 General topology of complexes
Full Text: DOI
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