Spectral $\zeta$-invariants lifted to coverings
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- by Sara Azzali and Sylvie Paycha PDF
- Trans. Amer. Math. Soc. 373 (2020), 6185-6226 Request permission
Abstract:
The canonical trace and the Wodzicki residue on classical pseudodifferential operators on a closed manifold are characterised by their locality and shown to be preserved under lifting to the universal covering as a result of their local feature. As a consequence, we lift a class of spectral $\zeta$-invariants using lifted defect formulae which express discrepancies of $\zeta$-regularised traces in terms of Wodzicki residues. We derive Atiyah’s $L^2$-index theorem as an instance of the $\mathbb {Z}_2$-graded generalisation of the canonical lift of spectral $\zeta$-invariants and we show that certain lifted spectral $\zeta$-invariants for geometric operators are integrals of Pontryagin and Chern forms.References
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Additional Information
- Sara Azzali
- Affiliation: Fachbereich Mathematik, Analysis und Differentialgeometrie, Universität Hamburg, Bundesstrasse 55, D-20146 Hamburg, Germany
- MR Author ID: 932572
- Email: sara.azzali@uni-hamburg.de
- Sylvie Paycha
- Affiliation: Institute of Mathematics, Universität Potsdam, Campus II - Golm, Haus 9, Karl-Liebknecht-Straße 24-25, D-14476 Potsdam, Germany, (On leave from the Université Blaise Pascal, Clermont-Ferrand)
- MR Author ID: 137200
- Email: paycha@math.uni-potsdam.de
- Received by editor(s): December 17, 2017
- Received by editor(s) in revised form: December 14, 2019
- Published electronically: July 8, 2020
- Additional Notes: The first author acknowledges support by DFG grant Secondary invariants for foliations within the Priority Programme SPP 2026 “Geometry at Infinity”
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 6185-6226
- MSC (2010): Primary 47G30, 58J42, 58J40; Secondary 58J28, 19K56
- DOI: https://doi.org/10.1090/tran/8067
- MathSciNet review: 4155176