Spectral flow and intersection number (Q1261747)

Let \(F= F(\mathbb{H})\) be the set of bounded Fredholm operators on a separable Hilbert space \(\mathbb{H}\) of infinite dimension. A subset \(\mathfrak F\) of \(F\) consisting of selfadjoint operators has three components: \({\mathfrak F}= {\mathfrak F}_ + +{\mathfrak F}_ - +{\mathfrak F}_*\) (\({\mathfrak F}_ +({\mathfrak F}_ -)\) consists of essentially positive (negative) operators and \({\mathfrak F}_*\) consists of others). \({\mathfrak F}_*\) is the classifying space for the \(K^{-1}\)-group, especially there are the spectral flows: \(\pi_ 1({\mathfrak F}_*)\to\mathbb{Z}\). The Bott periodicity theorem for complex \(K\)-group says that the spectral flows and the Fredholm index are equivalent as topological invariants through the suspension or the desuspension. However, there are analytic difficulties in dealing with the spectral flows arising from the family of differential operators with different domains of definitions. In this paper the authors construct an embedding of the unitary group \(\gamma: U(N)\to{\mathfrak F}_*\) for each \(N\) satisfying some conditions and give a non-trivial example of the spectral flow arising from a variation of non-local elliptic boundary conditions imposed on a fixed elliptic differential operator and its spectral flow formula in terms of intersection numbers between certain singular cycles.