Hadamard matrices and their designs: a coding-theoretic approach

Abstract:

To every finite dimensional algebraic coefficient system (defined below) $(\Theta ,V)$ over the De Rham algebra $\Omega (M)$ of a manifold $M$, Sullivan builds a local system ${\rho _\Theta }:{\pi _1}(M) \to V$, in the topological sense, such that the two cohomologies $H_{{\rho _\Theta }}^{\ast }(M;V)$ and $H_\Theta ^{\ast }(\Omega (M);V)$ are isomorphic. In this paper, if ${\mathbf {K}}$ is a simplicial set and $(\Theta ,V)$ an algebraic system over the ${C^\infty }$ forms ${A_\infty }({\mathbf {K}})$, we prove a similar result. We use it to extend the Hirsch lemma to the case of fibration whose fiber is an Eilenberg-Mac Lane space with certain non nilpotent action of the fundamental group of the basis. We apply this to a model of the hyperbolic torus; different from the nilpotent one, this new model is a better mirror of the topology.