Prop of ribbon hypergraphs and strongly homotopy involutive Lie bialgebras

For any integer $d$ we introduce a prop $RHr{a}_d$ of oriented ribbon hypergraphs (in which “edges” can connect more than two vertices) and prove that it admits a canonical morphism of props, $$Holie{b}_d^{\diamond }⟶RHr{a}_d,$$ $Holie{b}_d^{\diamond }$ being the (degree shifted) minimal resolution of prop of involutive Lie bialgebras, which is non-trivial on every generator of $Holie{b}_d^{\diamond }$. We obtain two applications of this general construction. As a first application we show that for any graded vector space $W$ equipped with a family of cyclically (skew)symmetric higher products the associated vector space of cyclic words in elements of $W$ has a combinatorial $Holie{b}_d^{\diamond }$-structure. As an illustration we construct for each natural number $N\ge 1$ an explicit combinatorial strongly homotopy involutive Lie bialgebra structure on the vector space of cyclic words in $N$ graded letters which extends the well-known Schedler’s necklace Lie bialgebra structure from the formality theory of the Goldman-Turaev Lie bialgebra in genus zero. Second, we introduced new (in general, non-trivial) operations in string topology. Given any closed connected and simply connected manifold $M$ of dimension $\ge 4$. We show that the reduced equivariant homology ${\bar{H}}_{\bullet }^{S^1}(LM)$ of the space $LM$ of free loops in $M$ carries a canonical representation of the dg prop $Holie{b}_{2-n}^{\diamond }$ on ${\bar{H}}_{\bullet }^{S^1}(LM)$ controlled by four ribbon hypergraphs explicitly shown in this paper.