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On Invariants of Hirzebruch and Cheeger-Gromov. (English) Zbl 1037.57028
A basic question in the theory of manifolds reads as follows: is it true that two closed manifolds which are homotopy equivalent are also homeomorphic? Special cases of this question are the Poincaré and Borel Conjectures. Here the authors prove that if \(M\) is a compact oriented manifold of dimension \(4k+3\) with \(k>0\), such that the fundamental group \(\pi_1(M)\) is not torsion free, then there are infinitely many manifolds which are simply homotopy equivalent to \(M\), but not homeomorphic to \(M\). In order to detect this infinite number of manifolds, the authors construct a secondary invariant \(\tau_{(2)}\) on the structure set of \(M\), that conincides with the \(\rho\)-invariant of Cheeger-Gromov. Therefore, \(\rho\) is not a homotopy invariant for the manifolds in question. On the other hand, the authors prove that if \(\Gamma\) is a torsion-free discrete subgroup of \(SL_n(\mathbb{R})\), then \(\rho\) is a homotopy invariant for manifolds with fundamental group \(\Gamma\).

MSC:
57R67 Surgery obstructions, Wall groups
46L80 \(K\)-theory and operator algebras (including cyclic theory)
58J20 Index theory and related fixed-point theorems on manifolds
57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
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