Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Spectral $ \zeta$-invariants lifted to coverings


Authors: Sara Azzali and Sylvie Paycha
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
Published electronically: July 8, 2020
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 47G30, 58J42, 58J40, 58J28, 19K56

Retrieve articles in all journals with MSC (2010): 47G30, 58J42, 58J40, 58J28, 19K56


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

DOI: https://doi.org/10.1090/tran/8067
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”
Article copyright: © Copyright 2020 American Mathematical Society