This paper is a survey of much of the mathematics arising from the analytic study of higher signatures. A distinguishing feature of the paper is its emphasis on manifolds with boundary, both as objects of study in their own right and as pieces of cut-and-paste constructions of closed manifolds. This emphasis gives the eta form and higher eta invariants important roles. The exposition in this paper is clear but, of necessity, very concise; an extensive bibliography provides references for the details that cannot fit in even a lengthy survey.
The paper is organized along lines starting from three fundamental results: the oriented homotopy invariance of a \(4k\)-dimensional manifold’s \(L\)-genus (or signature), the cut-and-paste invariance of the same number, and, on a compact manifold with boundary, the homotopy invariance of the sum of the integral of the \(L\)-differential form and the boundary term known as the eta invariant. Brief discussions of various proofs of these results motivate proofs of properties of the higher signatures. The paper outlines some of the main approaches that have proven cases of the Novikov conjecture on the homotopy invariance of higher signatures. These approaches provide the framework within which to address higher signatures on manifolds with boundary. A discussion of conditions under which one can define higher signatures on manifolds with boundary and prove the homotopy invariance of these higher signatures motivates the definition of the higher eta invariants. The paper also outlines a counterexample to the cut-and-paste invariance of higher signatures on closed manifolds before discussing conditions under which this invariance holds.