Graded Poisson algebras on bordism groups of garlands

Abstract

Let Mm be an oriented manifold and let N be a set consisting of oriented closed manifolds of the same odd dimension n. We consider a topological space GN,M of commutative diagrams. Each commutative diagram consists of a few manifolds from N that are mapped to M and of one point spaces pt that are each mapped to a pair of manifolds from N.We consider the oriented bordism group Ω⁎(GN,M)=⊕i=0∞Ωi(GN,M). We introduce the operations ⋆ and [⋅,⋅] on Ω⁎(GN,M)⊗Q, that make Ω⁎(GN,M)⊗Q into a Z-graded (m−2n)-Poisson algebra.For N={S1} and a surface M=F2, the subalgebra Ω0(G{S1},F2)⊗Q of our algebra is related to the Andersen-Mattes-Reshetikhin Poisson algebra of chord-diagrams.