Deformation of string topology into homotopy skein modules

Abstract

Relations between the string topology of Chas and Sullivan and the homotopy skein modules of Hoste and Przytycki are studied. This provides new insight into the structure of homotopy skein modules and their meaning in the framework of quantum topology. Our results can be considered as weak extensions to all orientable 3-manifolds of classical results by Turaev and Goldman concerning intersection and skein theory on oriented surfaces. AMS Classification 57M25; 57M35, 57R42Keywords 3-manifold, string topology, deformation, skein module, torsion, link homotopy, free loop space, Lie algebra IntroductionIn 1999 Moira Chas and Dennis Sullivan discovered the structure of a graded Lie algebra on the equivariant homology of the free loop space of an oriented smooth or combinatorial d-manifold [2]. Later Cattaneo, Fröhlich and Pedrini developed ideas about the quantization of string topology in the framework of topological field theory [1]. It is the goal of this paper to study VassilievKontsevitch and skein theory of links in 3-manifolds in relation with string topology. Our approach is intrinsically 3-dimensional. It hints towards a general deformation theory for a category of modules over the Chas-Sullivan Lie bialgebra (see also [3]) of an oriented 3-manifold. But the line of thought will not follow the ideas of classical quantization as in [1].Here is the main idea of the paper: For M a 3-dimensional manifold, the Chas-Sullivan structure measures the oriented intersections between the loops in a 1-dimensional family with those in a 0-dimensional familiy (which is just a formal linear combination of free homotopy classes of loops). The resulting pairing takes values in formal linear combinations of free homotopy classes. It is easy to see that the construction extends to collections of loops. It is our Our main method is transversality of families, which is at the heart of Vassiliev theory [21]. We will show that the process of replacing homotopy classes of maps by equivalence classes of transverse objects (oriented links in the case of 0-dimensional homology of the mapping space) naturally deforms the ChasSullivan type intersection theory into well-known structures in quantum topology. Thus our approach provides a weakened version of quantization in the category of oriented 3-manifolds in comparison to the deep results for cylinders over oriented surfaces due to Goldman [5] and Turaev [20]. Because of the lack of a geometric product structure in an arbitrary oriented 3-manifold such an extension will not be based on the deformation of algebra structures but the structures of modules over the Chas-Sullivan Lie algebra.In order to define a natural refinement of the Chas-Sullivan structure, isotopy has to be weakened to link homotopy of oriented links in M . Recall that in a deformation through link homotopy arbitrary self-crossings are allowed but the different components have to be disjoint during a deformation. This equivalence relation has first been considered by Milnor in 1952 [16] (also see the recent ...

HOMFLYPT Skein Theory, String Topology and 2-Categories

“…Proof. This is just Chernov's result [11], see also the author's approach in [27]. It easily extends to the case k ≥ 1.…”

“…The first version of this article was written in 2003, following results by the author on homotopy skein modules [26], [27] and two-term skein modules of framed links [28]. The techniques used in this paper are extending the ideas of those papers and modifications of the original ideas of Lin [37], Kalfagianni [29], Kalfagianni-Lin [31] and Kirk-Livingston [35], who construct finite type invariants of links in 3-manifolds.…”

“…Proof. Most of the arguments are already contained in [26] and [27]. Fix a component map α of the space map[0] corresponding to α ∈ b[0].…”

“…In the topological case, which we consider here, the passage from q = 1 to Jones theory is more easily described by introducing a local system on the space imm. This viewpoint has already been used in [27]. We refer to there and to [52] for some of the technical details.…”

“…and then apply Lemma 8.1. Note that S f is isomorphic to the quotient of RL t by the submodules generated by elements q −1 K (+) − K and q −1 K + − qK − (see [27]). Remark 8.2.…”