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In the middle of the last century, after hearing a talk of Mostow on one of his rigidity theorems, Borel conjectured in a letter to Serre a purely topological version of rigidity for aspherical manifolds (i.e. manifolds with contractible universal covers). The Borel conjecture is now one of the central problems of topology with many implications for manifolds that need not be aspherical. Since then, the theory of rigidity has vastly expanded in both precision and scope. This book rethinks the implications of accepting his heuristic as a source of ideas. Doing so leads to many variants of the original conjecture - some true, some false, and some that remain conjectural. The author explores this collection of ideas, following them where they lead whether into rigidity theory in its differential geometric and representation theoretic forms, or geometric group theory, metric geometry, global analysis, algebraic geometry, K-theory, or controlled topology.
The theorem of Novikov, that the rational Pontrjagin classes of a smooth manifold axe invariant under homeomorphisms, was a landmark in the development of the topology of manifolds. The geometric techniques introduced by Novikov were built upon by Kirby and Siebenmann in their study of topological manifolds. At the same time the problem was posed by Singer of developing an analytical proof of Novikov's original theorem.
The first such analytic proof was given by Sullivan and Teleman, building on deep geometric results of Sullivan which showed the existence and uniqueness of Lipschitz structures on high-dimensional manifolds. (It is now known that this result is false in dimension 4 – see.) However, the geometric techniques needed to prove Sullivan's theorem are at least as powerful as those in Novikov's original proof. For this reason, the Sullivan-Teleman argument (and the variants of it that have recently appeared) do not achieve the objective of replacing the geometry in Novikov's proof by analysis.
In an unpublished but widely circulated preprint, one of us (S.W.) suggested that this objective might be achieved by the employment of techniques from coarse geometry. A key part in the proposed proof is played by a certain homotopy invariance property of the ‘coarse analytic signature’ of a complete Riemannian manifold. We will explain in section 2 below what the coarse analytic signature is, in what sense it is conjectured to be homotopy invariant, and how Novikov's theorem should follow from the conjectured homotopy invariance. In section 3 we will prove the homotopy invariance modulo 2-torsion in the case that the control space is a cone on a finite polyhedron.