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Valuations on polyhedra and topological arrangements

Askold Khovanskii, Valentina Kiritchenko, Vladlen Timorin

TL;DR

This work develops a unified, functorial framework for simple $\mathbb{Z}^n$-valuations on polyhedra bounded by hyperplanes from a (possibly topological) collection $\hat{\mathcal{A}}$, linking scissors-congruence concepts to Hadwiger-type invariants. Central to the approach is the Hadwiger invariant calculus $\Upsilon_{\mathcal{L}}$, which together with reciprocity laws and period-vanishing conditions completely describes the space of valuations via a dual-flag, Leray-operator architecture. The authors extend classical results from affine hyperplane arrangements to topological arrangements, toric and pseudotoric settings, and pseudoaffine cases, using a robust chain- and cochain-based toolkit that unifies geometry, topology, and algebraic combinatorics. The main contributions include a precise description of simple valuations as generated by Hadwiger invariants, explicit relations among these generators, and a scalable framework that accommodates infinite sums and broad generalizations, with potential connections to toric geometry and algebraic K-theory. This framework enables systematic analysis of valuations in diverse geometric contexts and provides a foundation for future exploration of polynomial valuations and broader topological-geomtric structures.

Abstract

We revisit a classical theme of (general or translation invariant) valuations on convex polyhedra. Our setting generalizes the classical one, in a ``dual'' direction to previously considered generalizations: while previous research was mostly concerned with variations of ground fields/rings, over which the vertices of polytopes are defined, we consider more general collections of defining hyperplanes. No algebraic structures are imposed on these collections. This setting allows us to uncover a close relationship between scissors congruence problems on the one hand and finite hyperplane arrangements on the other hand: there are many parallel results in these fields, for which the parallelism seems to have gone unnoticed. In particular, certain properties of the Varchenko--Gelfand algebras for arrangements translate to properties of polytope rings or valuations. Studying these properties is possible in a general topological setting, that is, in the context of the so-called topological arrangements, where hyperplanes do not have to be straight and may even have nontrivial topology.

Valuations on polyhedra and topological arrangements

TL;DR

This work develops a unified, functorial framework for simple -valuations on polyhedra bounded by hyperplanes from a (possibly topological) collection , linking scissors-congruence concepts to Hadwiger-type invariants. Central to the approach is the Hadwiger invariant calculus , which together with reciprocity laws and period-vanishing conditions completely describes the space of valuations via a dual-flag, Leray-operator architecture. The authors extend classical results from affine hyperplane arrangements to topological arrangements, toric and pseudotoric settings, and pseudoaffine cases, using a robust chain- and cochain-based toolkit that unifies geometry, topology, and algebraic combinatorics. The main contributions include a precise description of simple valuations as generated by Hadwiger invariants, explicit relations among these generators, and a scalable framework that accommodates infinite sums and broad generalizations, with potential connections to toric geometry and algebraic K-theory. This framework enables systematic analysis of valuations in diverse geometric contexts and provides a foundation for future exploration of polynomial valuations and broader topological-geomtric structures.

Abstract

We revisit a classical theme of (general or translation invariant) valuations on convex polyhedra. Our setting generalizes the classical one, in a ``dual'' direction to previously considered generalizations: while previous research was mostly concerned with variations of ground fields/rings, over which the vertices of polytopes are defined, we consider more general collections of defining hyperplanes. No algebraic structures are imposed on these collections. This setting allows us to uncover a close relationship between scissors congruence problems on the one hand and finite hyperplane arrangements on the other hand: there are many parallel results in these fields, for which the parallelism seems to have gone unnoticed. In particular, certain properties of the Varchenko--Gelfand algebras for arrangements translate to properties of polytope rings or valuations. Studying these properties is possible in a general topological setting, that is, in the context of the so-called topological arrangements, where hyperplanes do not have to be straight and may even have nontrivial topology.
Paper Structure (85 sections, 27 theorems, 35 equations)

This paper contains 85 sections, 27 theorems, 35 equations.

Table of Contents

  1. Historic overview
  2. Introduction and statement of results
  3. Examples
  4. Variations of the setup
  5. Generators and relations
  6. Functoriality
  7. A general paradigm
  8. Flags in the torus
  9. Hadwiger invariants
  10. Reciprocity laws
  11. Period vanishing conditions
  12. Main theorem
  13. All simple valuations
  14. Topological arrangements
  15. Organization of the paper
  16. ...and 70 more sections

Key Result

Theorem 1

The Hadwiger invariants $\Upsilon_\mathcal{L}$ generate the space $\mathrm{Val}(\mathcal{P}_{\hat{\mathcal{A}}},\mathbb{Z}^n)$: every $\mathbb{Z}^n$-invariant simple valuation $\mu$ on $\hat{\mathcal{A}}$-polytopes can be written as $\mu_f=\sum_{\mathcal{L}} f(\mathcal{L}) \Upsilon_\mathcal{L}$ with

Theorems & Definitions (48)

  • Theorem
  • Theorem
  • Theorem
  • Lemma
  • proof
  • Theorem
  • Lemma
  • proof
  • Theorem
  • proof
  • ...and 38 more