# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1812.04913