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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.
##### Keywords:
Signature; $$L^2$$-signature; structure set; $$\rho$$-invariant
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