Patrick Orson

The eta invariant was introduced by Atiyah, Patodi and Singer in 1973 with the motivation of extending Hirzebruch's signature theorem to a 4k-dimensional manifold with boundary. It has found applications in knot theory as, under certain conditions, the eta invariant (or rather its 'reduced' version) can be shown to give a smooth, or even homotopy invariant of a (4k-1)-dimensional manifold M.

By first taking a unitary representation of the fundamental group of our (4k-1)-manifold M we can construct a twisted (reduced) eta invariant. It is then natural to consider what happens to the eta invariant if we take a 1-parameter family of such representations. The answer is that this path of eta-invariants has integral jumps at discrete points - these jumps can be interpreted as spectral flow of the twisted signature operator used to define the eta invariant (i.e. as eigenvalues changing from positive to negative).

Farber and Levine showed that jumps in the eta invariant can also be interpreted in terms of local homological data on the manifold and this is the point of view we begin with. In the situation that M is a fibre bundle over the circle, the homological data can be viewed as linking (Blanchfield) data or equivalently as automorphism data on the intersection form of the fibre. I am developing a bigger L-group called the 'double L-group' to capture all signature information related to the jumps in the eta invariant. This will certainly have an interpretation in terms of double sliceness of knots.

By weaving 4 fibration sequences into a braid we can see the interrelations between the 3-dimensional spin and complex spin groups. This diagram helps to show when an oriented 3-dimensional real vector bundle will admit one of these structures.