  We compare the invariants of flat vector bundles defined by Atiyah et al. and
Jones et al. and prove that, up to weak homotopy, they induce the same map,
denoted by $e$, from the $0$-connective algebraic $K$-theory space of the
complex numbers to the homotopy fiber of the Chern character. We examine
homotopy properties of this map and its relation with other known invariants.
In addition, using the formula for $\tilde{\xi}$-invariants of lens spaces
derived from Donnelly's fixed point theorem and the $4$-dimensional cobordisms
constructed via relative Kirby diagrams, we recover the formula for the real
part of $e$-invariants of Seifert homology spheres given by Jones and Westbury,
up to sign.
  We conjecture that this geometrically defined map $e$ can be represented by
an infinite loop map. The results in its companion paper [Wang2] give strong
evidence for this conjecture.
