Graded Poisson Algebras on Bordism Groups of Garlands and Their Applications

Let M be an oriented manifold and let N be a set consisting of oriented closed manifolds of possibly different dimensions. Roughly speaking, the space GN,M of N-garlands in M is the space of mappings into M of singular manifolds obtained by gluing manifolds from N at some marked points. In our previous work with Rudyak we introduced a rich algebra structure on the bordism group Ω∗(GN,M ). In this work we introduce the operations ⋆ and [·, ·] on the tensor product of Q and a certain bordism group Ω̂∗(GN,M ). For N consisting of odd-dimensional manifolds, these operations make Ω̂∗(GN,M ) ⊗ Q into a graded Poisson algebra (Gerstenhaber-like algebra). For N consisting of even-dimensional manifolds, ⋆ satisfies a graded Leibniz rule with respect to [·, ·], but [·, ·] does not satisfy a graded Jacobi identity. The mod 2-analogue of [·, ·] for one-element sets N was previously constructed in our preprint with Rudyak. For N = {S} and a surface F , the subalgebra Ω̂0(G{S1},F2) ⊗ Q of our algebra is related to the Goldman-Turaev algebra of loops on a surface and to the Andersen-Mattes-Reshetikhin Poisson algebra of chord-diagrams. As an application, our Lie bracket allows one to compute the minimal number of intersection points of loops in two given homotopy classes δ̂1, δ̂2 of free loops on F , provided that δ̂1, δ̂2 do not contain powers of the same element of π1(F ). 1. Basic definitions and main results In this paper the word “smooth” means C, and pt denotes the one-point space. For a smooth manifold M and a compact smooth manifold N we write C(N,M) for the standard topological space of smooth maps N → M. (When N is compact the strong and the weak topologies on the space of smooth maps N → M coincide.) We write C(N,M) for the topological space of continuous maps N → M. For a set X and an abelian ring R we write RX for the free R-module over the set X. Fix an oriented connected smooth manifold M of dimension m and fix a set N of pairwise different oriented connected smooth closed manifolds, possibly of different dimensions. Below we define the space GN,M that we call the space of N-garlands in M . Essentially it is the topological space of finite commutative diagrams that look as follows. Each diagram consists of M , a finite number of copies of pt-spaces, and a finite number of manifolds N ∈ N. We do allow the manifolds from N to participate more than once in a diagram. All the N -manifolds in a diagram are enumerated with the enumeration starting from 1, and we denote by Ni the manifold enumerated by i in a given diagram. Each manifold Ni in a diagram is smoothly mapped to M by exactly one map. Each pt-space in a diagram is mapped to some nonzero number of Ni-manifolds. We do allow diagrams such that for some of the Ni-manifolds in them the diagram contains no map from pt to these Ni. To a pt-space in such a diagram we correspond its multi index I which is the ordered sequence of indices i of manifolds Ni to which the pt is mapped. For example if pt is mapped to N1, N3, N5 then its multi index is {1, 3, 5}. Each such commutative diagram D gives rise to the oriented graph Γ(D) with vertices denoted by M , ptI , Ni and with a manifold from N associated to each vertex Ni. (It may be useful to think of the manifold from N associated to a vertex Ni as a color of the vertex.) Note that the resulting graph is rather special, for example there are no directed edges starting at M, or directed edges from N to pt . For simplicity of exposition, in this work we consider only the “tree-like” commutative diagrams, ie the commutative diagrams as above for which the nonoriented graph Γ(D) obtained from Γ(D) by forgetting the orientation on all the edges and deleting the M -vertex (together with all the edges leading to it) is a disjoint union of tree graphs. We also prohibit commutative diagrams for which one of the connected components of Γ(D) is a star-shape graph consisting of exactly one N -vertex and more than one pt-vertices. (Due to a