Two years ago my student Daniel Cibotaru wrote a dissertation entitled

Localization formulae in odd K-theory

in which he answers precisely this question in great generality.

More precisely, given a smooth family $(T_b)_{b\in B}$ of complex, Fredholm selfadjoint operators parametrized by a compact, connected, oriented smooth manifold $B$ he describes explicitely a (stratified) cycle in $B$ that is Poincare dual to the odd Chern character of this family. This cycle is non-homogeneous, i.e., it is a sum of cycles of various codimensions. The codimension $1$-part is the so called **Maslov class** or **spectral flow class** and it is more or less known. Daniel has a very nice description of the codimension $3$-part. For example if $B$ is a $3$-manifold, then the family defines a cohomology class in $H^3(B,\mathbb{Z})\cong \mathbb{Z}$ and Daniel explains how to compute this as a signed count of points in $B$.

More precisely, for a generic family $(T_b)_{b\in B}$, the locus of points $b$ where $\ker T_b\neq 0$ is an oriented surface $S\subset B$ and the family of vector spaces

$$ S\ni s\mapsto \ker T_s $$

is a complex line bundle over $S$.The above integer is none other than the degree of this complex line bundle.

The higher codimension parts have explicit but rather complicated descriptions.

The philosophy of his dissertation can be easily described: he constructs a smooth model for the classifying space of $K^1$. This is an infinite dimensional Grassmanian-like object $\mathcal{X}$ equipped with a Schubert-like stratification by strata of finite codimensions. He then shows that the closures of the Schubert strata determine cohomology classes forming an integral basis of the cohomology of $\mathcal{X}$. The whole thing has a strong symplectic flavor.