Dirac index classes and the noncommutative spectral flow
type: Item
instance of: scholarly article
title: Dirac index classes and the noncommutative spectral flow
zbMATH Open document ID: 1030.58018
DOI: 10.1016/S0022-1236(02)00044-7
published in: Journal of Functional Analysis
publication date: 2003-07-01T00:00:00Z
review text: The evaluation on \([M]\) of Hirzebruch's \(L\) polynomial in the Pontryagin classes of a smooth oriented closed manifold \(M\) equals the topological signature \(\sigma(M)\), and so it defines an oriented-homotopy invariant. When \(M\) is not simply connected, ``higher'' signatures can be defined by multiplying \(L\) with the pull-back of a class from \(B\pi_1(M)\). The Novikov conjecture asserts that these higher signatures are homotopy invariants. In the paper under review, Leichtnam and Piazza consider more generally maps \(f:M\to B\Gamma\), where \(B\Gamma\) is the classifying space of a discrete group \(\Gamma\), and Dirac operators on \(M\) twisted with the induced flat bundle \({\mathcal V}_f\), whose fibers are modeled on the reduced \(C^*\)-algebra \(\Lambda\) of the group \(\Gamma\). These Dirac operators are a particular case of the construction of \textit{A.~S.~Mishchenko} and \textit{A.~T.~Fomenko} [Math. USSR, Izv. 15, 87-112 (1980); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 43, 831-859 (1979; Zbl 0416.46052)]. The authors extend to the higher setting the notion of spectral section of \textit{R.~B.~Melrose} and \textit{P.~Piazza} [J. Differ. Geom. 46, No.1, 99-180 (1997; Zbl 0955.58020)], acknowledging earlier unpublished results of F. Wu. This result forms the technical core of the paper. It is needed in the definition of the higher Atiyah-Patodi-Singer- (APS-) and \(b\)-index classes for a manifold with boundary. These two classes are shown to coincide; as a corollary, the difference of the APS-index classes for two spectral projections \({\mathcal P}\) and \({\mathcal P'}\) is shown to equal \([{\mathcal P}-{\mathcal P'}]\in K_0(\Lambda)\). The authors also extend to the higher setting the notion of spectral flow, and give a splitting formula for the index. The final results concern the generalization of the cut-and-paste defect formula for the index of \textit{B.~Booss-Bavnek} and \textit{K.~Wojchiechowski} [C. R. Math. Acad. Sci., Soc. R. …
reviewed by: Sergiu Moroianu
Mathematics Subject Classification ID: 58J30
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 46L87
zbMATH DE Number: 1944979
zbMATH Keywords: nocommutative spectral projection
zbMATH Keywords: boundary value problems
zbMATH Keywords: gluing formulae
zbMATH Keywords: cut-and-paste invariance
zbMATH Keywords: Atiyah-Patodi-Singer index
zbMATH Keywords: \(b\)-index classes
zbMATH Keywords: spectral flow
author: Eric Leichtnam
author: Paolo Piazza
MaRDI profile type: Publication
cites work: Q4222027
cites work: Q4002484
cites work: Q4274286
cites work: On the gluing problem for the \(\eta\)-invariant
cites work: Q5482622
cites work: Cyclic cohomology, the Novikov conjecture and hyperbolic groups
cites work: Splitting of the family index
cites work: Higher spectral flow
cites work: Type II Spectral Flow and the Eta Invariant
cites work: Q5674136
cites work: On the homotopy invariance of higher signatures for manifolds with boundary
cites work: On the cut-and-paste property of higher signatures of a closed oriented manifold
cites work: The $b$-pseudodifferential calculus on Galois coverings and a higher Atiyah-Patodi-Singer index theorem
cites work: Higher Atiyah-Patodi-Singer index theory on Galois coverings
cites work: Spectral sections and higher Atiyah-Patodi-Singer index theory on Galois coverings
cites work: Homotopy invariance of twisted higher signatures on manifolds with boundary
cites work: A higher Atiyah-Patodi-Singer index theorem for the signature operator on Galois coverings
cites work: On higher eta-invariants and metrics of positive scalar curvature
cites work: Superconnections and higher index theory
cites work: Higher eta-invariants
cites work: Signatures and higher signatures of \(S^1\)-quotients
cites work: Q4217454
cites work: Q4278406
cites work: Families of Dirac operators, boundaries and the \(B\)-calculus
cites work: An index theorem for families of Dirac operators on odd-dimensional manifolds with boundary
cites work: Manifold cutting and pasting groups
cites work: Q4382029
cites work: Q3135819
cites work: Q4379462
full work available at URL: s0022-1236(02)00044-7
OpenAlex ID: W2098179009
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description: scientific article; zbMATH DE number 1944979