Not all Vinnicombe metric neighbourhoods are homotopically connected

Abstract

We prove by counterexample that even for two transfer functions which are close in the Nu-gap metric of Vinnicombe (1993, 1999), there does not necessarily exist a Vinnicombe metric homotopy from one transfer function to the other, such that intermediate transfer functions in the homotopy remain close to the transfer function at the beginning of the homotopy. This implies that the Vinnicombe metric neighbourhoods of some transfer functions in L-infinity space, are not connected.