Graded Poisson algebras on bordism groups of garlands

Abstract

Let $M$ be an oriented manifold and let $\mathfrak{N}$ be a set consisting of oriented closed manifolds of the same odd dimension. We consider the topological space $G_{\mathfrak{N},M}$ of commutative diagrams. Each commutative diagram consists of a few manifolds from $\mathfrak{N}$ that are mapped to $M$ and a few one point spaces $pt$ that are each mapped to a pair of manifolds from $\mathfrak{N}$. We consider the oriented bordism group ${\Omega }_{\ast }(G_{\mathfrak{N},M})={\oplus }_{i=0}^{\infty }{\Omega }_i(G_{\mathfrak{N},M}).$ We introduce the operations $\star$ and $[\cdot ,\cdot ]$ on ${\Omega }_{\ast }(G_{\mathfrak{N},M})\otimes \mathbb{Q},$ that make ${\Omega }_{\ast }(G_{\mathfrak{N},M})\otimes \mathbb{Q}$ into a graded Poisson algebra (Gerstenhaber-like algebra). For $\mathfrak{N}=\{S^1\}$ and a surface M=$F^2,$ the subalgebra ${\Omega }_0(G_{\{S^1\},F^2})\otimes \mathbb{Q}$ of our algebra is related to the Andersen-Mattes-Reshetikhin Poisson algebra of chord-diagrams.

40 pages, 3 figures. The exposition is significantly improved. More references added and the technical proofs similar to the ones give are deleted referring an interested reader to the previous versions of this preprint

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