Graded Poisson algebras on bordism groups of garlands

英文摘要：

Let $M$ be an oriented manifold and let $\frak N$ be a set consisting of oriented closed manifolds of the same odd dimension. We consider the topological space $G_{\frak N, M}$ of commutative diagrams. Each commutative diagram consists of a few manifolds from $\frak N$ that are mapped to $M$ and a few one point spaces $pt$ that are each mapped to a pair of manifolds from $\frak N$. We consider the oriented bordism group $Ω_*(G_{\frak N, M})=\oplus_{i=0}^{\infty} Ω_i(G_{\frak N, M}).$ We introduce the operations $\star$ and $[\cdot, \cdot]$ on $Ω_*(G_{\frak N,M})\otimes \mathbb Q,$ that make $Ω_*(G_{\frak N,M})\otimes \mathbb Q$ into a graded Poisson algebra (Gerstenhaber-like algebra). For ${\frak N}=\{S^1\}$ and a surface M=$F^2,$ the subalgebra $Ω_0(G_{\{S^1\}, F^2})\otimes \mathbb Q$ of our algebra is related to the Andersen-Mattes-Reshetikhin Poisson algebra of chord-diagrams.