Prop of ribbon hypergraphs and strongly homotopy involutive Lie bialgebras

TL;DR

This paper introduces a new prop of oriented ribbon hypergraphs and demonstrates its applications in constructing homotopy involutive Lie bialgebra structures and operations in string topology, linking algebraic and topological frameworks.

Contribution

It defines the prop RHra_d of ribbon hypergraphs, establishes a morphism from Holieb_d^ullet, and applies this to construct new algebraic structures in string topology and cyclic words.

Findings

Constructed a canonical morphism of props from Holieb_d^ullet to RHra_d.

Developed explicit strongly homotopy involutive Lie bialgebra structures on cyclic words.

Introduced new operations in string topology for manifolds of dimension ≥ 4.

Abstract

For any integer $d$ we introduce a prop $RHra_d$ of oriented ribbon hypergraphs (in which "edges" can connect more than two vertices) and prove that it admits a canonical morphism of props, $$ Holieb_d^\diamond \longrightarrow RHra_d, $$ $Holieb_d^\diamond$ being the (degree shifted) minimal resolution of prop of involutive Lie bialgebras, which is non-trivial on every generator of $Holieb_d^\diamond$. 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 $Holieb_d^\diamond$-structure. As an illustration we construct for each natural number $N\geq 1$ 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 $\geq 4$. 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 $Holieb_{2-n}^\diamond$ on $\bar{H}_\bullet^{S^1}(LM)$ controlled by four ribbon hypergraphs explicitly shown in this paper.