2022/08/12 by Taylor, Laurence R.
#11T24 (Primary) 57R67 #58K65 (Secondary) #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2208.06319
We consider Gauss sums associated to functions T→ \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 Poincaré 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 Poincaré spaces with boundary. Finally we apply our results to the bilinear form Sq1x∪ y on H1(M;\mathbb Z/2\mathbb Z) of an orientable 3-manifold.