Gauss Sums in Algebra and Topology

We consider Gauss sums associated to functions T → R/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 enable us to give elementary proofs of many standard results. We derive the Milgram Gauss sum formula computing the signature mod 8 of a non– singular bilinear form over Q and its generalization to non–even lattices. We generalize formulae of Brown on the signature mod 8 of non–singular integral forms and a generalization of it due to Kirby and Melvin. These results follow with no additional analysis and require no results on Witt groups. Assuming a bit of algebraic topology, we reprove a theorem of Morita’s computing the signature mod 8 of an oriented Poincaré duality space from the Pontryagin 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 Poincaré spaces with boundary. We apply our results to the bilinear form Sqx∪y on H(M ;Z/2Z) of an orientable 3–manifold and also derive Levine’s formula for the Arf invariant of a knot.