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Mapping surgery to analysis. I: Analytic signatures. (English) Zbl 1083.19002
In this first of a series of three papers [for part II, see ibid. 33, 301–324 (2004; Zbl 1083.19003)], the authors develop the theory of analytically controlled Poncaré complexes over a \(C^{*}\)-algebra \(C\). The main feature in this paper is the definition of a signature of such a complex, with values in the \(K\)-theory of the \(C^{*}\)-algebra \(C\). The setup is as follows, let \(C\) be a \(C^{*}\)-algebra, and \(E_{0}@<b_{1}<<E_{1}@<b_{2}<< \cdots@<b_{n}<< E_{n}\) a complex of Hilbert \(C\)-modules, where the differentials \(b_{i}\) are bounded, adjoinable operators. Moreover, an \(n\)-dimensional Hilbert-Poincaré complex \((E,b,T)\), over \(C\) is a complex of finitely generated Hilbert modules \(E_{a}@<b<<E_{a+1}@<b<< \cdots @<b<<E_{n-a}\) and adjoinable operators \(T:E_{p}\to E_{n-p}\) such that
(1) if \(v\in E_{p}\) then \(T^{*}v=(-1)^{(n-p)p}Tv\);
(2) if \(v\in E_{p}\) then \(Tb^{*}v+(-1)^{p}bTv=0\); and
(3) \(T\) induces an isomorphism from the homology of the dual complex above to the homology of the complex \((E,b)\).
Next, given an \(n\)-dimensional Hilbert-Poincaré complex \((E,b,T)\), let \(S:E\to E\) be the operator defined as \(Sv=i^{p(p-1)+l}Tv\) where \(n=2l\) or \(n=2l+1\) and \(i=\sqrt{-1}\), and let \(B=b^{*}-b\), then the signature of \((E,b,T)\) is defined as the class in \(K_{1}(C)\) of the operator \((B+S)(B-S)^{-1}:E_{\text{even}}\to E_{\text{even}}\) when \((E,b,T)\) is odd dimensional and the class in \(K_{0}(C)\) of \([P_{+}]-[P_{-}]\) of the positive projections of \(B+S\) and \(B-S\) if \(n\) is even. The authors prove that this signature is homotopy invariant, and that it coincides with the usual signature in special cases. The authors carry this signature to analytically controlled Hilbert-Poincaré complexes and prove that this is homotopy invariant and bordism invariant as well.

19J25 Surgery obstructions (\(K\)-theoretic aspects)
19K99 \(K\)-theory and operator algebras
Full Text: DOI
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