Cochain operations and subspace arrangements (Q1395907)

The author gives a combinatorial description of higher-order cohomology operations on mod \(p\) singular cohomology. The technique is to apply the classicial observation that the \(R\)-module \(C^m(X;R)\) of normalized singular cochains of degree \(m\) on a simpicial set \(X\) is isomorphic to the \(R\)-module of morphisms of simplicial sets \[ X \longrightarrow C^m(\Delta^\bullet;R) \] where \(\Delta^\bullet = \{\Delta^n\}\) is the standard cosimplicial space made out of the simplicial simplices. Let \(m\) vary yields a universal example to study. The main result, then, is that over a finite field, all cochain operations are polynomials in the face operators.