Graded Poisson algebras on bordism groups of garlands and their applications

Details Graded Poisson algebras on bordism groups of garlands and their applications Graded Poisson algebras on bordism groups of garlands and their applications

2006-08-06

arxiv.org/abs/math/0608153v3

Mathematics

Geometric Topology

64 pages, 5 figures, Finally the algebras are defined on the tensor product of Q and the oriented bordism group. (Previously w

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Graded Poisson algebras on bordism groups of garlands and their applications does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Graded Poisson algebras on bordism groups of garlands and their applications, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Graded Poisson algebras on bordism groups of garlands and their applications will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-601453