The aim is to develop in detail the theory of a new spectral invariant which we call the homological index and to fi?nd examples where it is non-trivial. We believe that we have to develop an homology theory and dual cohomology theory and pair them to produce a function on the cyclic homology of a certain algebra. This would generalise results in Kaad's thesis and also relate directly to the Witten index studied in the 90s. The homological index is detecting geometric information that is more subtle than can be understood using conventional index theory.
To understand the geometric content we study Dirac type operators that generalise those considered in the 80s and exist in all dimensions (not just one and two dimensions as in earlier work).
There is a map from the unbounded Dirac theory to the bounded operator theory used for the homological index. This map will be understood in more depth and used to tie together the spectral invariants of the unbounded operators with the homological invariants.