1-loop graphs and configuration space integral for embedding spaces
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. Phys. 256, No. 3, 513--537 (2005; Zbl 1101.57012)]. These are graphs whose first Betti number is 1, and the first author has shown in [loc. cit.] that graphs whose first Betti number is zero can also produce nontrivial classes. The latter are elements that detect the first non-trivial homotopy class of \(\text{Emb}(\mathbb R^j, \mathbb R^n)\) when \(n-j\) is odd, but the classes produced in this paper are of higher degrees and appear to be new. As the authors remark, a nice generalization would be to extend these non-triviality results to graphs with an arbitrary number of loops (the first author has defined and used such graphs in the paper mentioned above).NEWLINENEWLINEThe precise statement of the main theorem of the paper is as follows: With \(\mathcal A_k\) a vector space spanned by certain graphs (and with \(k\geq 2\) such that this space is non-trivial; the authors give conditions for this as well),NEWLINENEWLINE1) the group \(\text H^{(n-j-2)k}(\text {Emb}(\mathbb R^j, \mathbb R^n); \mathcal A_k)\) is non-trivial if: (i) \(n\) is odd; or (ii) \(n\) is even, \(j\) is odd, and \(k\leq 4\); or (iii) \(n\geq 12\) is even and \(j=3\); or (iv) \(n\) and \(j\) are even and \(2k(n-j-2)>j(2n-3j-3)\).NEWLINENEWLINE2) the group \(\text H^{(n-j-2)k}(\overline{\text {Emb}}(\mathbb R^j, \mathbb R^n); \mathcal A_k)\) is non-trivial if \(n\) and \(j\) are even.NEWLINENEWLINEThe reason for the difference in the statements for embeddings and embeddings modulo immersions is that, in some cases, the authors could only handle the case of ``anomalous faces'' in the latter setup (to show a form is closed, one needs to check the vanishing of the configuration space integral along all the faces of a certain compactified configuration space; the anomalous faces often gets in the way in this kind of a situation).NEWLINENEWLINEThe results in this paper are useful not only because they produce new cohomology classes of spaces of long embeddings but also since they generalize other classes constructed by others (Cattaneo-Rossi and Rossi in the papers mentioned above as well as \textit{T. Watanabe}, [``Configuration space integral for long \(n\)-knots and the Alexander polynomial'', Algebr. Geom. Topol. 7, 47--92 (2007; Zbl 1133.57016)]). In addition, the work here complements that of \textit{G. Arone} and \textit{V. Turchin} who show in [``On the rational homology of high dimensional analogues of spaces of long knots'', \url{arXiv:1105.1576v3}], using homotopy-theoretic methods, that the homology of \(\text{Emb}(\mathbb R^j, \mathbb R^n)\) is given by a certain graph complex for \(n\geq 2j+2\).
scientific article; zbMATH DE number 6033595