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The signature operator at 2. (English) Zbl 1103.58012

Let \(M\) be a closed smooth \(n\)-manifold. The Kasparov theory associates a \(K\)-homology class to any elliptic differential operator on \(M\). In particular, if \(M\) is oriented and endowed with a Riemannian metric, the signature operator determines a \(K\)-homology class \(\Delta_M\), which is independent of the metric (it is a diffeomorphism invariant). We have \(\Delta_M\in K_0(M)\) when \(n\) is even, and \(\Delta_M\in K_1(M)\) when \(n\) is odd; thus \(\Delta_M\in K_n(M)\) by Bott periodicity. This class \(\Delta_M\) has been studied by many authors after inverting \(2\) (in \(K_n(M)[1/2]\)), obtaining that it essentially agrees with Sullivan’s \(K[1/2]\)-orientation of topological manifolds. On the contrary, the purpose of the authors is to study \(\Delta_M\) localized at \(2\). The main results show that \(\Delta_M\) is an oriented bordism invariant and its reduction mod \(8\) is an oriented homotopy invariant, but its reduction mod \(16\) is not an oriented homotopy invariant (the lens space of dimension five provides a counterexample).

MSC:

58J20 Index theory and related fixed-point theorems on manifolds
19K35 Kasparov theory (\(KK\)-theory)
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