  Let $A(t)$ be an elliptic, product-type suspended (which is to say
parameter-dependant in a symbolic way) family of pseudodifferential operators
on the fibres of a fibration $\phi$ with base $Y.$ The standard example is
$A+it$ where $A$ is a family, in the usual sense, of first order, self-adjoint
and elliptic pseudodifferential operators and $t\in\bbR$ is the `suspending'
parameter. Let $\pi_{\cA}:\cA(\phi)\longrightarrow Y$ be the
infinite-dimensional bundle with fibre at $y\in Y$ consisting of the
Schwartz-smoothing perturbations, $q,$ making $A_y(t)+q(t)$ invertible for all
$t\in\bbR.$ The total eta form, $\eta_{\cA},$ as described here, is an even
form on $\cA(\phi)$ which has basic differential which is an explicit
representative of the odd Chern character of the index of the family: %
d\eta_{\cA}=\pi_{\cA}^*\gamma_A, \Ch(\ind(A))=[\gamma_{A}]\in H^{\odd}(Y).
\tag{*}\label{efatoi.5} % The 1-form part of this identity may be interpreted
in terms of the $\tau$ invariant (exponentiated eta invariant) as the
determinant of the family. The 2-form part of the eta form may be interpreted
as a B-field on the K-theory gerbe for the family $A$ with \eqref{efatoi.5}
giving the `curving' as the 3-form part of the Chern character of the index. We
also give `universal' versions of these constructions over a classifying space
for odd K-theory. For Dirac-type operators, we relate $\eta_{\cA}$ with the
Bismut-Cheeger eta form.
