A functorial LMO invariant for Lagrangian cobordisms

Abstract

Lagrangian cobordisms are three-dimensional compact oriented cobordisms between once-punctured surfaces, subject to some homological conditions. We extend the Le-Murakami-Ohtsuki invariant of homology three-spheres to a functor from the category of Lagrangian cobordisms to a certain category of Jacobi diagrams. We prove some properties of this functorial LMO invariant, including its universality among rational finite-type invariants of Lagrangian cobordisms. Finally, we apply the LMO functor to the study of homology cylinders from the point of view of their finite-type invariants. 57M27; 57M25 IntroductionThe Kontsevich integral is an invariant of links in S 3 , the standard 3-sphere. In their papers [23; 24], Le and Murakami extended this invariant to a functor from the category of tangles in the standard cube OE 1; 1 3 to the category of Jacobi diagrams based on 1-manifolds. One of the main features of the Kontsevich integral is its universality among rational-valued finite-type invariants of tangles (in the Goussarov-Vassiliev sense).Le, Murakami and Ohtsuki constructed in [26] an invariant of closed oriented 3-manifolds, which is called the Le-Murakami-Ohtsuki invariant. The LMO invariant is defined from the Kontsevich integral via surgery presentations of 3-manifolds in S 3 . For rational homology 3-spheres, the LMO invariant is universal among rational-valued finite-type invariants (in the Ohtsuki sense). Later, Murakami and Ohtsuki extended in [34] the LMO invariant to an invariant of 3-manifolds with boundary, which satisfies modified axioms of TQFT. More recently, Le and the first author constructed from the LMO invariant a functor from a certain category of 3-dimensional cobordisms to a certain category of modules [6]. Let us recall that each of those two constructions [34; 6] (ii) Unify the extended Kontsevich integral and the LMO invariant into a single invariant Z.M; G/ of couples .M; G/, where M is a closed oriented 3-manifold and G M is an embedded framed trivalent graph.Then, a compact oriented 3-manifold with boundary is obtained from each couple .M; G/ by cutting a regular neighborhood N.G/ of G in M . If the connected components of G were split into two parts, say the "top" part G C and the "bottom" part G , then M n N.G/ can be regarded as a cobordism between closed surfaces, namely from @N.G C / to @N.G /. Finally, the LMO invariant of the cobordism M n N.G/ is defined in [34; 6] to be the Kontsevich-LMO invariant Z.M; G/ of the couple .M; G/.In this paper, we propose an alternative solution to the problem of extending the LMO invariant to 3-manifolds with boundary. In contrast with the previous two constructions [34; 6], we prefer to work with cobordisms between once-punctured surfaces. This technical choice has two advantages: On the one hand, it avoids extending the Kontsevich integral to trivalent graphs in S 3 ; on the other hand, it allows us to work with monoidal categories, and to construct tensor-preserving functors.Moreover, we normalize the Kontsevich-LMO invariant Z to obtain ...

Symplectic Jacobi diagrams and the Lie algebra of homology cylinders

“…As proposed in [14] and established in [4], there is a diagrammatic version of the Lie algebra Gr Y C g,1 ⊗ Q. Similar diagrammatic constructions have also been considered by Garoufalidis and Levine [9] and by Habegger [12].…”

“…It is shown in [4] that, if an M ∈ C g,1 is Y i -equivalent to the trivial cylinder, then Z Y (M ) − ∅ starts in internal degree i. So, the LMO map induces a multiplicative map…”

“…In this section, we recall from [4,14] the main ingredients to obtain Theorem 1.2, which gives a diagrammatic description of the Lie algebra of homology cylinders. Furthermore, we produce from the LMO invariant a diagrammatric description of the Malcev Lie algebra of the group of homology cylinders.…”

“…with values in a certain complete Hopf algebra of Jacobi diagrams. The latter is isomorphic via a certain map ϕ defined in [4] to A < (−H Q ). Thus, the composition s • ϕ • Z Y defines a monoid homomorphism…”

Johnson-Morita theory in mapping class groups and monoids of homology cobordisms of surfaces

“…Habiro-Massuyeau [38] determined the structure of G Y C g,1 ⊗Q by using the LMO functor defined in Cheptea-Habiro-Massuyeau [15]. See also a paper by Andersen, Bene, Meilhan and Penner [2] for a related work.…”