Greenberg, Peter A smooth scissors congruence problem. (English) Zbl 0539.57012 Proc. Am. Math. Soc. 89, 298-302 (1983). 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 Cited in 2 Documents 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 Keywords:classifying spaces of codimension 2 foliations; 1-dimensional non- Hausdorff manifold; scissors congruence problem; Hilbert’s third problem; homology fibration theorem for actions of groupoids Citations:Zbl 0398.57018 PDFBibTeX XMLCite \textit{P. Greenberg}, Proc. Am. Math. Soc. 89, 298--302 (1983; Zbl 0539.57012) Full Text: DOI References: [1] P. Greenberg, A model for groupoids of homeomorphisms, Thesis, M.I.T., Cambridge, Mass., 1982. [2] -, Extension and restriction for manifolds, preprint, 1982. [3] A. Haefliger, Feuilletages sur les variétés ouvertes, Topology 9 (1970), 183 – 194 (French). · Zbl 0196.26901 · doi:10.1016/0040-9383(70)90040-6 [4] André Haefliger, Homotopy and integrability, Manifolds – Amsterdam 1970 (Proc. Nuffic Summer School), Lecture Notes in Mathematics, Vol. 197, Springer, Berlin, 1971, pp. 133 – 163. [5] John N. Mather, Integrability in codimension 1, Comment. Math. Helv. 48 (1973), 195 – 233. · Zbl 0284.57016 · doi:10.1007/BF02566122 [6] Dusa McDuff, On groups of volume-preserving diffeomorphisms and foliations with transverse volume form, Proc. London Math. Soc. (3) 43 (1981), no. 2, 295 – 320. · Zbl 0411.57028 · doi:10.1112/plms/s3-43.2.295 [7] J. Palis and S. Smale, Structural stability theorems, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 223 – 231. · Zbl 0214.50702 [8] C.-H. Sah, Hilbert’s third problem: scissors congruence, Pitman, London, 1979. · Zbl 0406.52004 [9] Graeme Segal, Categories and cohomology theories, Topology 13 (1974), 293 – 312. · Zbl 0284.55016 · doi:10.1016/0040-9383(74)90022-6 [10] Graeme Segal, Classifying spaces related to foliations, Topology 17 (1978), no. 4, 367 – 382. · Zbl 0398.57018 · doi:10.1016/0040-9383(78)90004-6 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.