Invariants of three-dimensional manifolds
type: Item
instance of: scholarly article
title: Invariants of three-dimensional manifolds
zbMATH Open document ID: 0870.57017
DOI: 10.1007/BF02247888
author: Sergei Gelfand
author: David Kazhdan
published in: Geometric and Functional Analysis. GAFA
publication date: 1997-08-31T00:00:00Z
full work available at URL: https://eudml.org/doc/58227
review text: The authors consider the combinatorial approach to the construction of invariants of a three-dimensional manifold and construct an invariant starting from a monoidal category \(\mathbf C\) without any braiding conditions, i.e commutativity conditions of the form \(X\otimes Y\simeq Y\otimes X\). For any three-dimensional manifold \(M\) with boundary \(S\) and a triangulation \(D\) of \(M\), they construct a finite-dimensional linear space \(W(S,D')\) depending on the restriction \(D'\) of \(D\) to \(S\) (a triangulation of \(S\)) and a vector \(I_{\mathbf C}(M,S,D)\in W(S,D')\). The main result of the paper is that \(I_{\mathbf C}(M,S,D)\) depends only on \(D'\) and not on \(D\) itself. Taking the inductive limit of spaces \(W(S,D')\) over all triangulations \(D'\) of \(S\), the authors construct a space \(K(S)\) and a vector \(I_{\mathbf C}(M,S)\in K(S)\), which is an invariant of a three-dimensional manifold \(M\) with boundary \(S\). In particular, when \(M\) is a closed manifold, \(S=\emptyset\), they have \(K(\emptyset)=k\), and their construction gives an invariant \(I_{\mathbf C}(M)\in k\).
reviewed by: Vyacheslav Abramov
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 18D10
zbMATH DE Number: 912454
zbMATH Keywords: invariants
zbMATH Keywords: monoidal category
zbMATH Keywords: triangulation
zbMATH Keywords: 3-manifold
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 912454