A Combinatorial Method for Computing Steenrod Squares

Details A Combinatorial Method for Computing Steenrod Squares A Combinatorial Method for Computing Steenrod Squares

2001-10-29

arxiv.org/abs/math/0110308v1

Journal of Pure and Applied Algebra 139 (1999) 89-108

Mathematics

Algebraic Topology

25 pages

Scientific paper

10.1016/S0022-4049(99)00006-7

We present here a combinatorial method for computing cup-$i$ products and Steenrod squares of a simplicial set $X$. This method is essentially based on the determination of explicit formulae for the component morphisms of a higher diagonal approximation (i.e., a family of morphisms measuring the lack of commutativity of the cup product on the cochain level) in terms of face operators of $X$. A generalization of this method to Steenrod reduced powers is sketched. This description can be considered as a translation of the most ancient definition of Steenrod squares to the general setting of the Simplicial Topology.

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