We present a detailed proof of the existence-theorem for
noncommutative spectral sections (see the noncommutative spectral flow, unpublished preprint, 1997). We apply this result to various index-theoretic situations, extending to the noncommutative context results of Booss–Wojciechowski, Melrose–Piazza and Dai–Zhang. In particular, we prove a variational formula, in
K*(
Cr*(Γ)), for the index classes associated to 1-parameter family of Dirac operators on a Γ-covering with boundary; this formula involves a
noncommutative spectral flow for the boundary family. Next, we establish an additivity result, in
K*(
Cr*(Γ)), for the index class defined by a Dirac-type operator associated to a closed manifold
M and a map
r:
M→
BΓ when we assume that
M is the union along a hypersurface
F of two manifolds with boundary
M=
M+ F M−. Finally, we prove a
defect formula for the signature-index classes of two cut-and-paste equivalent pairs (
M1,
r1:
M1→
BΓ) and (
M2,
r2:
M2→
BΓ), where
M1=M+ (F,φ1) M−, M2=M+ (F,φ2) M−
and φ
jDiff(
F). The formula involves the noncommutative spectral flow of a suitable 1-parameter family of twisted signature operators on
F. We give applications to the problem of cut-and-paste invariance of Novikov's higher signatures on closed oriented manifolds.
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