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Rigidity and crystallographic groups. I. (English) Zbl 0692.57017
Rigidity results in topology originated from the work of L. Bieberbach [see 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., F. T. Farrel and W. C. Hsiang, Am. J. Math. 105, 641-672 (1983; Zbl 0521.57018), and F. T. Farrell and L. E. Jones, Invent. Math. 91, 559-586 (1988; Zbl 0657.57015); cf. W. C. Hsiang and 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 F. Conolly and 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 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.
Reviewer: K.Pawałowski

MSC:
57S30 Discontinuous groups of transformations
57R85 Equivariant cobordism
53C20 Global Riemannian geometry, including pinching
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
57R80 \(h\)- and \(s\)-cobordism
57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
57R19 Algebraic topology on manifolds and differential topology
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