LAUSR

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 $G_{N, M}$ 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 oriented bordism group $\bor_*(G_{N,M}).$ In this work we introduce the operations $\star$ and $[.,.]$ on $\bor_*(G_{N,M})\otimes Q.$ For $N$ consisting of odd-dimensional manifolds, these operations make $\bor_*(G_{N,M})\otimes Q$ into a graded Poisson algebra (Gerstenhaber-like algebra). For $N$ consisting of even-dimensional manifolds, $\star$ 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 $N={S^1}$ and a surface $F^2,$ the subalgebra 0-dimensional bordism group subalgebra of our algebra is related to the Goldman-Turaev and to the Andersen-Mattes-Reshetikhin Poisson algebras. As an application, our Lie bracket allows one to compute the minimal number of intersection points of loops in two given homotopy classes $\delta_1, \delta_2$ of free loops on $F^2,$ that do not contain powers of the same loop.