nLab scissors congruence

Idea

Two polygons $P,P'$ in the Euclidean plane have the same area iff they are scissors congruent in the sense that they can be subdivided into finitely many pieces such that the pieces of $P$ are congruent to the pieces of $P'$.

Hilbert’s third problem (see Hilbert's problems) was to decide whether the analogue of this elementary fact holds for polyhedra in 3-dimensional space. Dehn solved this problem using what is now called the Dehn invariant. In modern language, it assigns to a polyhedron $P$ an element in $\mathbf{R}\otimes_{\mathbf{Z}} \mathbf{R}/\mathbf{Z}$. If both the Dehn invariant and volume of two (3-dimensional finite) polyhedra in Euclidean space are equal then they are scissors congruent.

The generalized Hilbert’s third problem asks whether two finite $n$-polytopes in Euclidean, spherical or hyperbolic $n$-space must be scissors equivalent if they have the same volume and generalized invariants.

The scissors congruence group $\mathcal{P}(X,G)$ where $G$ is a subgroup of the group of isometries of $X$, is the free abelian group on symbols $[P]$, for all polytopes in $X$ modulo the relations

(i) $[P]-[P']-[P'']$ when $P = P'\coprod P''$,

(ii) $[g P]-[P]$.

In the book

• J.L. Dupont, Scissors congruences, group homology and characteristic classes (.ps) , Lecture notes from Nankai Institute of Mathematics, 1998, Manuscript, University of Aarhus, Department of Math., Aarhus, 1999, 122 pp., and in: Nankai Tracts in Mathematics 1, World Scientific, Singapore, 2001.

Dupont explains the relations of dilogarithms, Reidemeister torsion, Dehn invariant (which is from 1900) and their role in understanding the 19th century scissors congruence…The subject is very active now. After physicist Anatole Kirillov found the quantum dilogarithm and L. D. Fadeeev and Kashaev published a paper about it (and discovered the pentagon relation), this became an important topic in quantization (Fadeev’s modular double, quantization of Teichmuller spaces, work of A. Fock, L. Takhdajan, Aldrovandi etc.) of hyperbolic spaces and generally the hyperbolic geometry and with relations, together with classical dilogarithm to many other subjects (e.g. to the work of Reznikov and theory or regulators, specially Borel regulator in higher algebraic K-theory, important for study of motives (Goncharov et al.) and, as W. Nahm shown, also relevant for understanding some phenomena in CFT).

References

Scissor congruence

• C. H. Sah, Hilbert’s third problem: scissors congruence, Research Notes in Mathematics 33, Pitman 1979.

• Inna Zakharevich, Scissors congruence as K-theory, arxiv/1101.3833

• J.-P. Sydler, Conditions nécessaires et suffisantes pour l’équivalence des polyèdres de l’espace euclidean à trois dimensions, Comment. Math. Helv. 40, 43-80, 1965.

• Wikipedia, Hilbert’s third problem

Cut-and-paste of manifolds

Generalization to cut-and-paste of smooth manifolds (SK-groups, D: Schneiden-und-Kleben Gruppen):

• U. Karras, Matthias Kreck, Walter D. Neumann, E. Ossa, Cutting and Pasting of Manifolds; SK-Groups, Publish or Perish (1973) [pdf, pdf]

• Walter D. Neumann, Manifold cutting and pasting groups, Topology 14 3 (1975) 237-244 [doi:10.1016/0040-9383(75)90004-X]

Relation to TQFTs:

• Carmen Rovi, Matthew Schoenbauer, Relating Cut and Paste Invariants and TQFTs, The Quarterly Journal of Mathematics 73 2 (2022) 579–607 [arXiv:1803.02939, doi:10.1093/qmath/haab044]

Survey:

• Carmen Rovi, Relating cut and paste invariants and TQFTS, talk at CQTS (Apr. 2023) [video: YT]