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Dehn-Sydler-Jessen Via Homological Algebra

Anubhav Nanavaty

TL;DR

The paper reframes Euclidean scissors congruence in terms of group homology with coefficients in the polytope module Pt(E^n) and the Tits complex, enabling a homological proof of the Dehn–Sydler–Jessen theorem. It develops the translational setting via G-scissors congruence, computes the relevant group-homology groups, and derives the DSJ exact sequence with the Dehn invariant D capturing the three-dimensional obstruction. In dimension three, the authors establish injectivity of the Dehn invariant and identify the cokernel with the space of K"ahler differentials Ω^1_R, using exceptional isomorphisms involving Spin, Pin, and Hochschild homology to relate geometric data (edge lengths and dihedral angles) to algebraic invariants. This approach clarifies the link between geometry and homology, providing a coherent, self-contained homological proof of DSJ and illuminating the structure underlying Hadwiger-type invariants in Euclidean 3-space.

Abstract

We provide an expository introduction to Euclidean Scissors Congruence, the study of polytopes in Euclidean space up to `cut and paste' relations. We first re-frame questions in scissors congruence as those in group homology. We then use this perspective to review the proof of the Dehn-Sydler-Jessen Theorem as found in the works of Dupont and Sah.

Dehn-Sydler-Jessen Via Homological Algebra

TL;DR

The paper reframes Euclidean scissors congruence in terms of group homology with coefficients in the polytope module Pt(E^n) and the Tits complex, enabling a homological proof of the Dehn–Sydler–Jessen theorem. It develops the translational setting via G-scissors congruence, computes the relevant group-homology groups, and derives the DSJ exact sequence with the Dehn invariant D capturing the three-dimensional obstruction. In dimension three, the authors establish injectivity of the Dehn invariant and identify the cokernel with the space of K"ahler differentials Ω^1_R, using exceptional isomorphisms involving Spin, Pin, and Hochschild homology to relate geometric data (edge lengths and dihedral angles) to algebraic invariants. This approach clarifies the link between geometry and homology, providing a coherent, self-contained homological proof of DSJ and illuminating the structure underlying Hadwiger-type invariants in Euclidean 3-space.

Abstract

We provide an expository introduction to Euclidean Scissors Congruence, the study of polytopes in Euclidean space up to `cut and paste' relations. We first re-frame questions in scissors congruence as those in group homology. We then use this perspective to review the proof of the Dehn-Sydler-Jessen Theorem as found in the works of Dupont and Sah.

Paper Structure

This paper contains 6 sections, 31 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. The Polytope Module and the Tits Complex
  3. Translational Scissors Congruence
  4. The Case of $\mathbb{E}^3$: First Theorems
  5. The Case of $\mathbb{E}^3$: Exceptional Isomorphisms
  6. The group homology of $O(n)$ and $E(n)$

Key Result

Theorem 1.1

There is an exact sequence of groups: where

Theorems & Definitions (65)

  • Theorem 1.1: Dehn--Sydler--Jessen
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • ...and 55 more