Graphical composition of mapping spaces between modules of configuration-space-type

TL;DR

This paper develops a graphical composition framework for embedding spaces related to configuration spaces, providing explicit combinatorial models and describing the action of coinduction on certain modules, advancing the understanding of their rational homotopy types.

Contribution

It introduces a graphical composition method for embedding spaces and describes the coinduction action on configuration-space modules, bypassing complex algebraic models.

Findings

Graphical composition for embedding spaces $ ilde{Emb}$ constructed.

Explicit description of coinduction action on configuration modules.

Advances in rational homotopy type analysis of embedding spaces.

Abstract

In embedding calculus, spaces of embeddings are identified with derived mapping spaces between framed Fulton-MacPherson-type modules (framed configuration spaces). Unfortunately, there are no sufficiently good algebraic models for framed Fulton-MacPherson modules that would allow us to explicitly describe the rational homotopy type of the embedding space. Recently there were several attempts to avoid dealing with the framed versions of Fulton-MacPherson modules by considering framed manifolds, e.g. embeddings modulo immersions $\overline{\mathrm{Emb}}$ in a recent paper by Fresse, Turchin and Willwacher, or embeddings with a deformation of the framing $\widetilde{\mathrm{Emb}}$ in a recent paper by the author. In both cases, the rational homotopy type of the corresponding embedding space has an explicit description in terms of graphs (hairy graph complexes). We construct a combinatorial graphical composition for the composition of embedding spaces $\widetilde{\mathrm{Emb}}$. As a step on our way, we describe the action of the coinduction functor on configuration-space-type $e_n^c$-comodules.