Topological $K$-theory with coefficients and the $e$-invariant

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.