Hadamard matrices and their designs: a coding-theoretic approach

Abstract

To every finite dimensional algebraic coefficient system (defined below) ( Θ , V ) (\Theta ,V) over the De Rham algebra Ω ( M ) \Omega (M) of a manifold M M , Sullivan builds a local system ρ Θ : π 1 ( M ) → V {\rho _\Theta }:{\pi _1}(M) \to V , in the topological sense, such that the two cohomologies H ρ Θ ∗ ( M ; V ) H_{{\rho _\Theta }}^{\ast }(M;V) and H Θ ∗ ( Ω ( M ) ; V ) H_\Theta ^{\ast }(\Omega (M);V) are isomorphic. In this paper, if K {\mathbf {K}} is a simplicial set and ( Θ , V ) (\Theta ,V) an algebraic system over the C ∞ {C^\infty } forms A ∞ ( K ) {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.