  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.
