Configuration spaces and Vassiliev classes in any dimension

Abstract

The real cohomology of the space of imbeddings of S^1 into R^n, n>3, is studied by using configuration space integrals. Nontrivial classes are explicitly constructed. As a by-product, we prove the nontriviality of certain cycles of imbeddings obtained by blowing up transversal double points in immersions. These cohomology classes generalize in a nontrivial way the Vassiliev knot invariants. Other nontrivial classes are constructed by considering the restriction of classes defined on the corresponding spaces of immersions.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-39.abs.htm

A homotopy-theoretic view of Bott–Taubes integrals and knot spaces

“…This result concerned one particular knot invariant previously found through Chern-Simons theory. However, the framework they set up was used by Thurston [22] to construct a whole class of knot invariants for knots in R 3 (see also Volić [23]), and by Cattaneo, Cotta-Ramusino and Longoni [12] to construct cohomology classes in Emb.S 1 ; R n /. The knot invariants constructed in [22] are Vassiliev invariants (ie, finite type), and the graph cochain complex used in [12] to construct the cohomology classes is known to be quasi-isomorphic to the E 1 term of the Vassiliev spectral sequence.…”

Homology of spaces of knots in any dimensions

“…Moreover, the indices that invariants and cohomology classes de ne at all such essential strata (by appropriate non-obvious analogues of conditions of type f ( ) + f ( ) = f ( ) from the theory of knot invariants) are, in most situations, not scalar: they take values in certain homology groups associated with these strata (see Vassiliev 1999a). Therefore, the elementary interpretation of the¯ltration, as well as the very de¯nition of objects of¯nite type, should be adequately modi¯ed in any particular theory of this sort (see Cattaneo et al 2000;Vassiliev 1994bVassiliev , 1999d. Notice, however, the beautiful theory of nite-type invariants of 3-manifolds begun by Ohtsuki and extended by Garoufalidis, Goussarov and others (see Garoufalidis 1996;Goussarov 1999;Ohtsuki 1996).…”

Higher-Dimensional BF Theories¶in the Batalin-Vilkovisky Formalism:¶The BV Action and Generalized Wilson Loops

“…These were recently introduced in [16] thanks to a superformalism that simplifies a lot the combinatorics of the associated Batalin-Vilkovisky (BV) cohomology. The meaning of these observables is that their expectations values are cohomology classes on the space of framed imbeddings of S 1 (as those described in [13]). …”

“…Second, the quantum BV formalism requires considering the so-called BV Laplacian (see subsection 4.4) and this forces one to restrict to imbeddings (more precisely, to framed imbeddings). Third, one needs restrictions on the Lie algebra underlying the definition of the BF theory; a semisimple Lie algebra (or more generally a Lie algebra as in Assumption 3 on page 4) will do only in the odddimensional case and for a specific observable whose expectation value involves Feynman diagrams that, apart from an obvious dependence of the propagators on the dimension, are exactly the same as in the computation of knot invariants from Chern-Simons theory (see [13] and references therein); the main characteristic of these diagrams is that they involve completely antisymmetric trivalent vertices which satisfy a diagrammatic version of the Jacobi identity (see [5]): namely, each vertex represents a binary operation with the same properties of a Lie bracket. More general observables (and, in particular, any observable in the even-dimensional case) requires stronger restrictions on the involved Lie algebra; in particular, our construction works when it comes from an associative algebra (see Assumption 4 on page 36).…”

“…The Feynman diagrams for the expectation values of these generalized Wilson loops contain (k + 1)-valent vertices (for any k ≥ 2) which can be interpreted as k-ary operations satisfying certain conditions. The way BF theories generate vertices (i.e., k + 1-ary operations) is through one single binary operation (i.e., an associative product) plus a trace; but in principle there might exist more general algebraic structures generating graph cocycles (in the sense of [29] and made precise in [13]) which yield cohomology classes of imbedded circles.…”