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Steenrod operations via higher Bruhat orders

Guillaume Laplante-Anfossi, Nicholas J. Williams

TL;DR

The work provides a conceptual bridge between higher Bruhat orders and Steenrod cup-$i$ coproducts by embedding coproduct data into zonotopal tilings (cubillages) of cyclic zonotopes. It constructs a family of coproducts $ abla_i^U$ indexed by elements $U$ of the higher Bruhat order, proving they realize homotopies between $ abla_{i-1}$ and its opposite and that every coproduct satisfying the homotopy formula arises from such a cubillage. The framework yields a unified, geometric, and combinatorial account of Steenrod operations, extends to simplicial and singular contexts, and demonstrates the independence of Steenrod squares from the chosen coproduct via reoriented higher Bruhat orders. Together, these results connect Steenrod theory with simplex equations and cubical orientals, offering new perspectives and potential computational advantages for cohomology operations.

Abstract

The purpose of this paper is to establish a correspondence between the higher Bruhat orders of Yu. I. Manin and V. Schechtman, and the cup-$i$ coproducts defining Steenrod squares in cohomology. To any element of the higher Bruhat orders we associate a coproduct, recovering Steenrod's original ones from extremal elements in these orders. Defining this correspondence involves interpreting the coproducts geometrically in terms of zonotopal tilings, which allows us to give conceptual proofs of their properties and show that all reasonable coproducts arise from our construction.

Steenrod operations via higher Bruhat orders

TL;DR

The work provides a conceptual bridge between higher Bruhat orders and Steenrod cup- coproducts by embedding coproduct data into zonotopal tilings (cubillages) of cyclic zonotopes. It constructs a family of coproducts indexed by elements of the higher Bruhat order, proving they realize homotopies between and its opposite and that every coproduct satisfying the homotopy formula arises from such a cubillage. The framework yields a unified, geometric, and combinatorial account of Steenrod operations, extends to simplicial and singular contexts, and demonstrates the independence of Steenrod squares from the chosen coproduct via reoriented higher Bruhat orders. Together, these results connect Steenrod theory with simplex equations and cubical orientals, offering new perspectives and potential computational advantages for cohomology operations.

Abstract

The purpose of this paper is to establish a correspondence between the higher Bruhat orders of Yu. I. Manin and V. Schechtman, and the cup- coproducts defining Steenrod squares in cohomology. To any element of the higher Bruhat orders we associate a coproduct, recovering Steenrod's original ones from extremal elements in these orders. Defining this correspondence involves interpreting the coproducts geometrically in terms of zonotopal tilings, which allows us to give conceptual proofs of their properties and show that all reasonable coproducts arise from our construction.
Paper Structure (20 sections, 26 theorems, 88 equations, 3 figures)

This paper contains 20 sections, 26 theorems, 88 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Steenrod operations
  4. Chain complexes
  5. Steenrod coalgebra
  6. Higher Bruhat orders
  7. Admissible orders
  8. Consistent sets
  9. Cubillages
  10. From consistent sets to cubillages
  11. Coproducts from cubillages
  12. Comparison to original Steenrod operations
  13. The homotopy formula
  14. Homotopies from covering relations
  15. Implications of the construction
  16. ...and 5 more sections

Key Result

Theorem 1

For every element $U \in \mathcal{B}([0, n], i + 1)$, there is a coproduct which gives a homotopy between $\Delta_{i - 1}$ and $\Delta_{i - 1}^\mathrm{op}$. If $U_{\min}$ and $U_{\max}$ are the maximal and minimal elements of $\mathcal{B}([0,n], i + 1)$, then $\{\Delta_{i}^{U_{\min}}, \Delta_{i}^{U_{\max}}\} = \{\Delta_{i}, \Delta_{i}^{\mathrm{op}}\}$. Moreover, every copr

Figures (3)

  • Figure 1: Cubes associated to some vertices and sets of generating vectors.
  • Figure 2: The upper and lower cubillages of $Z(3, 2)$, the associated $\Delta_{1}$ coproducts, and their boundaries
  • Figure :

Theorems & Definitions (59)

  • Theorem : \ref{['const']}, \ref{['lem:boundary=steenrod', 'thm:homotopy_formula', 'thm:all_coproducts']}
  • Remark 2.1
  • Example 2.2
  • Theorem 2.3: ms89
  • Proposition 2.4: t02 or dkk18
  • Proposition 2.5: t02
  • Proposition 3.1
  • proof
  • Proposition 3.3
  • proof
  • ...and 49 more