Rigidity and crystallographic groups. I
Rigidity results in topology originated from the work of L. Bieberbach [see \textit{L. S. Charlap}: Bieberbach groups and flat manifolds (1986; Zbl 0608.53001) for a transparent exposition of related results]. A related conjecture, due to A. Borel, asserts that any two isomorphic, torsion free, properly discontinuous, co-compact subgroups of \(Homeo({\mathbb{R}}^ n)\) must be conjugate. This conjecture has been verified in a number of special cases [see, e.g., \textit{F. T. Farrel} and \textit{W. C. Hsiang}, Am. J. Math. 105, 641-672 (1983; Zbl 0521.57018), and \textit{F. T. Farrell} and \textit{L. E. Jones}, Invent. Math. 91, 559-586 (1988; Zbl 0657.57015); cf. \textit{W. C. Hsiang} and \textit{J. L. Shaneson}, Topology of Manifolds, Proc. Univ. Georgia 1969, 18-51 (1971; Zbl 0288.57006)]. In all cases, the proofs use the ``h-cobordism rigidity'' result stating that the Whitehead group Wh(\(\Gamma)\) vanishes for a torsion free group \(\Gamma\). The main theorem of the paper under review provides a new kind of ``h- cobordism rigidity'' result to the effect that the Tate cohomology \(\hat H^*({\mathbb{Z}}/2{\mathbb{Z}};Wh_ G^{top,\rho}(M_{\Gamma}))\) vanishes, where \(Wh_ G^{top,\rho}(M_{\Gamma})\) is the Whitehead group of G-h- cobordisms of the flat torus \(M_{\Gamma}\) of a crystallographic group \(\Gamma\) with holonomy group G. In order to define the Tate cohomology, the authors consider a function \[ Wh_ G^{top,\rho}(M_{\Gamma})\to Wh_ G^{top,\rho}(M_{\Gamma}) \] and using \textit{F. Conolly} and \textit{W. Lück}'s ``The involution on the equivariant Whitehead group'' [K- Theory 3, No.2, 123-140 (1989)], they show that it is a group homomorphism and an involution. The authors observe that the main theorem implies that any crystallographic manifold, with odd order holonomy, which is h-cobordant and simple homotopy equivalent to the standard one is actually homeomorphic to it. They also announce that in the second part of their work [Rigidity and crystallographic groups. II (to appear)], they show that one can remove the word ``h-cobordant'' from the previous sentence. A generalized (somewhat vague) version of the Borel conjecture is due to \textit{F. Quinn} [Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 598-606 (1987; Zbl 0673.57017)]. The authors propose a sharpened form of the Borel-Quinn conjecture, and they prove it in a special case.
torsion free, properly discontinuous, co-compact subgroups of \(Homeo({\mathbb{R}}^ n)\)
Whitehead group of G-h-cobordisms of the flat torus
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