Spectral flow and the unbounded Kasparov product

We present a fairly general construction of unbounded representatives for the interior Kasparov product. As a main tool we develop a theory of ^C1-connections on operator *-modules; we do not require any smoothness assumptions; our σ-unitality assumptions are minimal. Furthermore, we use work of Kucerovsky and our recent Local Global Principle for regular operators in Hilbert ^{C *}-modules.As an application we show that the Spectral Flow Theorem and more generally the index theory of Dirac-Schrödinger operators can be nicely explained in terms of the interior Kasparov product.