Gauss Sums in Algebra and Topology

Abstract

We consider Gauss sums associated to functions $T\to \mathbb R/\mathbb Z$ which satisfy some sort of quadratic property and investigate their elementary properties. These properties and a Gauss sum formula from the nineteenth century due to Dirichlet give the Milgram Gauss sum formula computing the signature mod $8$ of a non-singular bilinear form over $\mathbb Q$. Brown derived some results on the signature mod 8 of non-singular integral forms. Kirby and Melvin gave a formula for a generalization of this invariant to possibly non-singular forms and we further generalize it here. The Milgram Gauss sum formula and these formulas allow us to reprove Brown's result without resort to Witt group calculations. Assuming a bit of algebraic topology, we reprove a theorem of Morita's computing the signature mod $8$ of an oriented Poincar\'e duality space from the Pontrjagin square without using Bockstein spectral sequences. Since we work with forms which may be singular, we also obtain a version of Morita's theorem for Poincar\'e spaces with boundary. Finally we apply our results to the bilinear form $Sq^1x\cup y$ on $H^1(M;\mathbb Z/2\mathbb Z)$ of an orientable 3-manifold.