Friday, February 3, 2012

1202.0272 (Moulay Tahar Benameur et al.)

Index type invariants for twisted signature complexes    [PDF]

Moulay Tahar Benameur, Varghese Mathai
For a closed, oriented, odd dimensional manifold X, we define the rho
invariant rho(X,E,H) for the twisted odd signature operator valued in a flat
hermitian vector bundle E, where H = \sum i^{j+1} H_{2j+1} is an odd-degree
closed differential form on X and H_{2j+1} is a real-valued differential form
of degree {2j+1}. We show that $\rho(X,\E,H)$ is independent of the choice of
metrics on X and E and of the representative H in the cohomology class [H]. We
establish some basic functorial properties of the twisted rho invariant. We
express the twisted eta invariant in terms of spectral flow and the usual eta
invariant. In particular, we get a simple expression for it on closed oriented
3-dimensional manifolds with a degree three flux form. A core technique used is
our analogue of the Atiyah-Patodi-Singer theorem, which we establish for the
twisted signature operator on a compact, oriented manifold with boundary.
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