Cochain Operations and Subspace Arrangements

Stephan Klaus

Abstract: We give a combinatorial description of simplicial cochain and cocycle operations mod$p$ by certain subspace arrangements over $\Z/p$. Then we show that any primary or higher cohomology operation mod$p$ has a description at the cochain level by polynomials of coface operators. We define an algebraic filtration on the space of cochain operations and give an explicit dimension formula for the space of cocyle operation of fixed algebraic degree. The filtration is also defined on Steenrod operations and leads to a new spectral sequence converging to the cohomology of Eilenberg--MacLane spaces mod$p$.

Keywords: cochain operation; cohomology operation; coface operator; cross-effect; arrangement; Eilenberg--MacLane space.

Classification (MSC2000): 18G30, 18G35, 18G40 , 52B30, 55P20, 55S05, 55S10, 55S20, 55T90, 55U10.

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