Spectral 𝜁-invariants lifted to coverings

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 ζ \zeta -invariants using lifted defect formulae which express discrepancies of ζ \zeta -regularised traces in terms of Wodzicki residues. We derive Atiyah’s L 2 L^2 -index theorem as an instance of the Z 2 \mathbb {Z}_2 -graded generalisation of the canonical lift of spectral ζ \zeta -invariants and we show that certain lifted spectral ζ \zeta -invariants for geometric operators are integrals of Pontryagin and Chern forms.