The Universal  -Invariant for Manifolds With Boundary

Abstract

We extend the theory of the universal η-invariant to the case of bordism groups of manifolds with boundaries. This allows the construction of secondary descendants of the universal η-invariant. We obtain an interpretation of Laures' f -invariant as an example of this general construction. As an aside we improve a recent result by Han-Zhang about the modularity of a certain formal power series of η-invariants.

Relative torsion and bordism classes of positive scalar curvature metrics on manifolds with boundary

“…The η-invariant of the spin Dirac operator of DM twisted with the latter bundle can be shown to be an invariant of the bordism class of the psc metric on M (using an argument similar to the one used in the proof of the previous theorem). This can be seen as a generalisation of the APS ρ-invariant associated to a pair of representations for manifolds with boundary and seems to be strongly related to the invariants considered by Bunke in [6], which have inspired our constuction. Now we show that the relative L 2 -ρ-invariant is equal to the L 2 η-invariant of a suitable Γ/Im(ϕ) Γ -cover of DM .…”

“…One can define an η-invariant with values in the abelianisation of N (π 1 (M )/Im(π 1 (N )) π1(M) ) and our L 2 ρinvariant is then obtained from the latter η-invariant using the canonical trace of N (π 1 (M )/Im(π 1 (N )) π1(M) ). Using a pair of finite-dimensional representations of π 1 (M ) agreeing on π 1 (N ), we can also define a relative version of the APS ρ-invariants and in this case our invariants seem to be stongly related to the invariants defined by Bunke in [6], which has inspired our construction.…”