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A smooth scissors congruence problem. (English) Zbl 0539.57012

The author considers the following variation of the scissors congruence problem (Hilbert’s third problem): Let \(M=\coprod(a,b)_ f/\sim\), where \(f:(a,b)\to {\mathbb{R}}^ 2\) is a smooth embedding and \((a,b)_ f\) is a copy of the open interval (a,b). Identify \(x\in(a,b)_ f\) with \(y\in(c,d)_ g\) if there exists a diffeomorphism h:\(U\to V\) from a neighborhood U of x onto a neighborhood V of y such that \(f=g{\mathbb{O}}h\) on U. M is a 1-dimensional non-Hausdorff \(C^{\infty}\)-manifold. Let \(H_ 0(\Gamma^{\infty};H_ 1M)\) (resp. \(H_ 0(\Gamma^{\Omega};H_ 1M))\) be the quotient of the first singular homology group \(H_ 1M\) of M by the subgroup generated by elements \((i^*g)_*b-b,\quad b\in im(H_ 1(i^{-1}U)\to H_ 1M),\) where g:\(U\to V\) is some orientation preserving diffeomorphism (resp. area and orientation preserving diffeomorphism) between open sets of \({\mathbb{R}}^ 2\) and where \(i^*g:i^{-1}(U)\to i^{-1}(V)\) is the induced map on the inverse image of U under the obvious immersion \(i:M\to {\mathbb{R}}^ 2\). The main result of the author states that \(H_ 0(\Gamma^{\infty};H_ 1M)={\mathbb{Z}}\) and \(H_ 0(\Gamma^{\Omega};H_ 1M)={\mathbb{Z}}\oplus {\mathbb{R}}.\) The proof uses classifying space techniques and variants of the homology fibration theorem for actions of groupoids [for actions of monoids see for example §2 of G. Segal, Topology 17, 367-382 (1978; Zbl 0398.57018)].
Reviewer: E.Vogt

MSC:

57R99 Differential topology
57R32 Classifying spaces for foliations; Gelfand-Fuks cohomology
58H10 Cohomology of classifying spaces for pseudogroup structures (Spencer, Gelfand-Fuks, etc.)
57R50 Differential topological aspects of diffeomorphisms

Citations:

Zbl 0398.57018
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References:

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