1-loop graphs and configuration space integral for embedding spaces
type: Item
instance of: scholarly article
title: 1-loop graphs and configuration space integral for embedding spaces
zbMATH Open document ID: 1243.57022
DOI: 10.1017/S0305004111000429
author: Keiichi Sakai
author: Tadayuki Watanabe
published in: Mathematical Proceedings of the Cambridge Philosophical Society
publication date: 2012-05-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/1002.4696
Mathematics Subject Classification ID: 57R40
Mathematics Subject Classification ID: 57R42
Mathematics Subject Classification ID: 57Q45
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 58D10
Mathematics Subject Classification ID: 55R80
zbMATH DE Number: 6033595
MaRDI profile type: Publication
arXiv ID: 1002.4696
OpenAlex ID: W3106443903
review text: This paper expands the work done by the first author in [``Configuration space integrals for embedding spaces and the Haefliger invariant'', J. Knot Theory Ramifications 19, No. 12, 1597--1644 (2010; Zbl 1223.57024)] and presents various non-trivial cohomology classes of the space of long embeddings of \(\mathbb R^{j}\) in \(\mathbb R^{n}\) and the related space of long embeddings modulo immersions.NEWLINENEWLINEMore precisely, let \(\text{Emb}(\mathbb R^j, \mathbb R^n)\) be the space of embeddings of \(\mathbb R^j\) in \( \mathbb R^n\) with fixed behavior outside the unit disk. Let \(\text{Imm}(\mathbb R^j, \mathbb R^n)\) be the space of immersions with the same behavior outside the disk and let \(\overline{\text{Emb}}(\mathbb R^j, \mathbb R^n)\) be the homotopy fiber of the inclusion \(\text{Emb}(\mathbb R^j, \mathbb R^n)\hookrightarrow\text{Imm}(\mathbb R^j, \mathbb R^n)\). The authors use configuration space integrals to produce nontrivial cocycles in the deRham cohomology of \(\text{Emb}(\mathbb R^j, \mathbb R^n)\) and \(\overline{\text{Emb}}(\mathbb R^j, \mathbb R^n)\). To prove non-triviality of these classes, the authors use a generalization of the ``resolution of crossings'' idea that was used in [\textit{A. S. Cattaneo, P. Cotta-Ramusino} and \textit{R. Longoni}, ``Configuration spaces and Vassiliev classes in any dimension'', Algebr. Geom. Topol. 2, 949--1000 (2002; Zbl 1029.57009)] to show non-triviality of some cohomology classes of classical long knots.NEWLINENEWLINEThe closed forms produced by the authors take values in a certain complex of \textit{1-loop graphs} which were previously used by the first author but have also appeared in [\textit{C. Rossi}, ``Invariants of higher-dimensional knots and topological quantum field theories'', Zürich: Univ. Zürich, Mathematisch-naturwissenschaftliche Fakultät. (2002; Zbl 1141.81331)] and [\textit{A. S. Cattaneo} and \textit{C. A. Rossi}, ``Wilson surfaces and higher dimensional knot invariants'', Commun. Math…
reviewed by: Ismar Volic
description: scientific article; zbMATH DE number 6033595