[Paper Review] On the rational homotopy type of embedding spaces of manifolds in $R^n$

Benoît Fresse, Victor Turchin+1 more|arXiv (Cornell University)|Aug 18, 2020

Homotopy and Cohomology in Algebraic Topology28 references4 citations

TL;DR

This paper determines the rational homotopy type of embedding spaces of manifolds in ℝⁿ for codimensions n ≥ m+2 by constructing an explicit $L_∞$-algebra of diagrams that computes the rational homotopy groups of the homotopy fiber of the inclusion Emb(L,ℝⁿ) → Imm(L,ℝⁿ). The key result is that the rational homotopy type of connected components is governed by hairy graph complexes associated to the cohomology of the manifold, extending previous results to lower codimensions and resolving a conjecture on string link spaces.

ABSTRACT

We study the spaces of embeddings of manifolds in a Euclidean space. More precisely we look at the homotopy fiber of the inclusion of these spaces to the spaces of immersions. As a main result we express the rational homotopy type of connected components of those embedding spaces through combinatorially defined $L_\infty$-algebras of diagrams.

Motivation & Objective

• To determine the rational homotopy type of the space of embeddings of a manifold L in ℝⁿ, particularly the homotopy fiber over the trivial embedding.
• To extend previous results on rational homotopy groups of embedding spaces from codimension ≥ 2m+1 to the broader range n ≥ m+2.
• To resolve a conjecture in [STT] that the rational homotopy groups of string link spaces modulo immersions are computed by hairy graph complexes in codimension ≥ max(mᵢ)+3.
• To provide a systematic framework using $L_\infty$-algebras of diagrams to describe the rational homotopy type of embedding spaces.
• To relate the rational homotopy groups to Maurer–Cartan elements in graph-complexes and construct Sullivan models for connected components.

Proposed method

• The authors model the rational homotopy type of the embedding space via an $L_\infty$-algebra of diagrams, denoted $\mathrm{HGC}_{\bar{H}^*(M_*),n}$, derived from the homotopy commutative structure of the cohomology of the manifold.
• They use the homotopy fiber of the inclusion Emb(L,ℝⁿ) → Imm(L,ℝⁿ) as the central object, which classifies embeddings that are trivial as immersions.
• The construction relies on replacing the manifold L by its open tubular neighborhood $N(L)$ or $N_\infty(L)$ for unbounded components, preserving weak homotopy type.
• The rational homotopy groups are computed via the Chevalley–Eilenberg cohomology of the twisted $L_\infty$-algebra $\mathrm{HGC}_{\bar{H}^*(M_*),n}^m$ for a Maurer–Cartan element m.
• Sullivan models for connected components are constructed using the cohomological Chevalley–Eilenberg complex of the positive-degree truncation of the twisted $L_\infty$-algebra.
• The method generalizes to non-compact manifolds by using $N_\infty(L)$, and the results are shown to be compatible with known classification results for 3-manifolds in ℝ⁶.

Experimental results

Research questions

• RQ1 How can the rational homotopy type of embedding spaces of manifolds in ℝⁿ be described for codimensions n ≥ m+2?
• RQ2 Can the conjecture that hairy graph complexes compute the rational homotopy groups of string link spaces modulo immersions in codimension ≥ max(mᵢ)+3 be proven?
• RQ3 What is the role of the $L_\infty$-algebra of diagrams in encoding the rational homotopy groups of embedding spaces?
• RQ4 How do Maurer–Cartan elements in the graph complex relate to isotopy classes of embeddings?
• RQ5 Can Sullivan models be constructed for connected components of embedding spaces using diagrammatic $L_\infty$-algebras?

Key findings

• The rational homotopy type of the homotopy fiber $\overline{\mathrm{Emb}}(L,\mathbb{R}^n)$ is computed by the $L_\infty$-algebra $\mathrm{HGC}_{\bar{H}^*(M_*),n}$, which is constructed from the cohomology of the manifold and the codimension n.
• The conjecture in [STT] that the rational homotopy groups of $\overline{\mathrm{Emb}}_\partial(\coprod \mathbb{R}^{m_i}, \mathbb{R}^n)$ are computed by hairy graph complexes in codimension ≥ max(mᵢ)+3 is confirmed.
• For compact manifolds, the rational homotopy groups are computed by the same $L_\infty$-algebra, with higher homotopy operations not affecting the computation due to graph complexity.
• The rational homotopy groups of connected components are isomorphic to the Chevalley–Eilenberg cohomology of the positive-degree truncation of the twisted $L_\infty$-algebra.
• Sullivan models for connected components are given by the cohomological Chevalley–Eilenberg complex of the twisted $L_\infty$-algebra, providing a full rational model for the embedding space.
• The framework applies uniformly to both bounded and unbounded manifolds, with $N_\infty(L)$ providing a homotopy-equivalent model for the unbounded case.