Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-08T13:22:49.805Z Has data issue: false hasContentIssue false
  • English
  • Français

Product formula for Atiyah-Patodi-Singer index classes and higher signatures

Published online by Cambridge University Press:  01 April 2010

Charlotte Wahl
Charlotte Wahl
Affiliation:
Leibniz-Arbeitsstelle Hannover, der Göttinger Akademie der Wissenschaften, Waterloostr. 8, 30169 Hannover, Germany, [email protected]
Get access

Abstract

We define generalized Atiyah-Patodi-Singer boundary conditions of product type for Dirac operators associated to C*-vector bundles on the product of a compact manifold with boundary and a closed manifold. We prove a product formula for the K-theoretic index classes, which we use to generalize the product formula for the topological signature to higher signatures.

Keywords

higher signaturesboundary value problemsKK-theoryindex theory
Type
Research Article
Information
Journal of K-Theory , Volume 6 , Issue 2 , October 2010 , pp. 285 - 337
Copyright
Copyright © ISOPP 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BGV96.Berline, N., Getzler, E. and Vergne, M., Heat Kernels and Dirac Operators (Grundlehren der mathematischen Wissenschaften 298), Springer-Verlag, 1996Google Scholar
B198.Blackadar, B., K-Theory for Operator Algebras (MSRI Publications 5), 2nd edition, Cambridge Univ. Press, 1998Google Scholar
BW93.Booss-Bavnbek, B. and Wojciechowski, K. P., Elliptic boundary problems for Dirac operators, Birkhäuser, 1993CrossRefGoogle Scholar
Bu95.Bunke, U., “A K-theoretic relative index theorem and Callias-type Dirac operators”, Math. Ann. 303 (1995), 241–279CrossRefGoogle Scholar
HS92.Hilsum, M. and Skandalis, G., ‘Invariance par homotopie de la signature à coefficients dans un fibré presque plat”, J. reine angew. Math. 423 (1992), 73–99Google Scholar
LLK02.Leichtnam, E., Lück, W. and Kreck, M., “On the cut-and-paste property of higher signatures of a closed oriented manifold”, Topology 41 (2002), 725–744CrossRefGoogle Scholar
LLP00.Leichtnam, E., Lott, J. and Piazza, P., “On the homotopy invariance of higher signatures for manifolds with boundary”, J. Diff. Geom. 54 (2000), 561–633Google Scholar
LP98.Leichtnam, E. and Piazza, P., “Spectral sections and higher Atiyah-Patodi-Singer index theory on Galois coverings”, Geom. Funct. Anal. 8 (1998), 17–58CrossRefGoogle Scholar
LP99.Leichtnam, E. and Piazza, P., “Homotopy invariance of twisted higher signatures on manifolds with boundary”, Bull. Soc. Math. Fr. 127 (1999), 307–331CrossRefGoogle Scholar
LP00.Leichtnam, E. and Piazza, P., “A higher Atiyah-Patodi-Singer index theorem for the signature operator on Galois coverings”, Ann. Global Anal. Geom. 18 (2000), 171–189CrossRefGoogle Scholar
LP03.Leichtnam, E. and Piazza, P., “Dirac index classes and the noncommutative spectral flow”, J. Funct. Anal. 200 (2003), 348–400CrossRefGoogle Scholar
LP04.Leichtnam, E. and Piazza, P., “Elliptic operators and higher signatures”, Ann. Inst. Fourier 54 (2004), 1197–1277CrossRefGoogle Scholar
Lo92.Lott, J., “Higher eta-invariants”, K-Theory 6 (1992), 191–233CrossRefGoogle Scholar
Lü02.Lück, W., L2-invariants: Theory and Applications to Geometry and K-theory (Ergebnisse der Mathematik und ihrer Grenzgebiete 44), Springer-Verlag, 2002CrossRefGoogle Scholar
MP97a.Melrose, R. B. and Piazza, P., “Families of Dirac operators, boundaries and the b-calculus” J. Diff. Geom. 46 (1997), 99–180Google Scholar
MP97b.Melrose, R. B. and Piazza, P., “An index theorem for families of Dirac operators on odd-dimensional manifolds with boundary” J. Diff. Geom. 46 (1997), 287–334Google Scholar
RS01.Rosenberg, J. and Stolz, S., “Metrics of positive scalar curvature and connections with surgery”, Surveys on surgery theory 2 (Ann. of Math. Stud. 149), Princeton Univ. Press, 2001, 353–386Google Scholar
S05.Schick, T., ‘L 2-index theorems, KK-theory, and connections”, New York J. Math. 11 (2005), 387–443Google Scholar
ST01.Solovyov, Y. P. and Troitsky, E. V., C*-algebras and elliptic operators in differential topology (Translations of Mathematical Monographs 192), American Mathematical Society, 2001Google Scholar
St.Stolz, S., Concordance classes of positive scalar curvature metrics, preprint http://www.nd.edu/~stolz/concordance.psGoogle Scholar
W07.Wahl, C., “On the noncommutative spectral flow”, J. Ramanujan Math. Soc. 22 (2007), 135–187Google Scholar
W09.Wahl, C., The Atiyah-Patodi-Singer index theorem for Dirac operators over C*-algebras, preprint arXiv:0901.0381Google Scholar
We99.Weinberger, S., “Higher ρ-invariants”, Tel Aviv topology conference: Rothenberg Festschrift, Contemp. Math. 231 (1999), 315–320CrossRefGoogle Scholar
Wu97.Wu, F., “The higher Γ-index for coverings of manifolds with boundaries”, Cyclic cohomology and noncommutative geometry (Fields Inst. Commun. 17), Amer. Math. Soc., 1997, 169–183Google Scholar