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Stabilization by free products giving rise to Andrews-Curtis equivalences. (English) Zbl 0781.57001
Note Mat. 10, Suppl. 2, 305-314 (1990).
Let \(K^ n\) and \(L^ n\) be compact, connected, \(n\)-dimensional CW- complexes. For \(n\neq 2\) it is known that \(K^ 2\curvearrowright L^ n\) implies \(K^ n\curvearrowright^{n+1}L^ n\), [C. T. C. Wall, Proc. Lond. Math. Soc., III. Ser. 16, 342-352 (1966; Zbl 0151.313)]. The generalized Andrews-Curtis conjecture asserts that this result is also true when \(n=2\). For a discussion of this and related questions see Chapters I and XII [both written by the authors in “Two-dimensional homotopy and combinatorial group theory” (edited by the authors with A. Sieradski), London Math. Soc. Lecture Notes Series 197 (1993)]. In the paper under review the following is shown. Let \(K^ 2_ 0\) and \(L^ 2_ 0\) be compact, connected, 2-dimensional CW-complexes which are simple-homotopy equivalent. Then for suitably large \(r\), the 1-point unions \(K^ 2=K^ 2_ 0\lor K^ 2_ 1\lor K^ 2_ 2\lor\cdots\lor K^ 2_ r\), \(L^ 2=L^ 2_ 0\lor K^ 2_ 1\lor K^ 2_ 2\lor\cdots\lor K^ 2_ r\), where the \(K^ 2_ i\) \((i=1,\ldots,r)\) are models of the presentation \(\langle x,y;x^ 2,[x,y],y^ 4\rangle\) of \(\mathbb{Z}_ 2\times\mathbb{Z}_ 4\), are such that \(K^ 2\curvearrowright^ 3L^ 2\). As the authors remark, if the \(K^ 2_ i\) \((i=1,\ldots,r)\) were replaced by an appropriate number of 2-spheres, then the theorem would hold under the weaker assumption that \(K^ 2_ 0\) and \(L^ 2_ 0\) have the same fundamental group and Euler characteristic (this is essentially Tietze’s theorem on changing presentations of groups). The point of the author’s theorem is that the \(K^ 2_ i\) \((i=1,\ldots,r)\) have minimal Euler characteristic for their fundamental group.

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