Séminaire Lotharingien de Combinatoire, 91B.10 (2024), 12 pp.
Guillaume Laplante-Anfossi and Nicholas J. Williams
From Higher Bruhat Orders to Steenrod Cup-i Coproducts
Abstract.
We show that the higher Bruhat orders of Manin and Schechtman provide
a useful conceptual framework for understanding Steenrod's cup-i
coproducts, which are used to define the cohomology operations known
as Steenrod squares.
Indeed, we show that the elements of the (i+1)-dimensional higher
Bruhat order are in bijection with all possible cup-i coproducts on
the chain complex of the simplex which give a homotopy between
cup-(i-1) and its opposite.
The Steenrod cup-i coproduct and its opposite are then given by the
maximal and minimal elements of the higher Bruhat order.
This correspondence uses the geometric realisation of the higher
Bruhat orders in terms of tilings of cyclic zonotopes, and enables us
to give conceptual proofs of the fundamental properties of the cup-i
coproducts.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
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