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Brunnian links and Kontsevich graph complex I

Boris Botvinnik, Tadayuki Watanabe

TL;DR

The paper constructs a natural chain map from the Kontsevich graph complex to the rational singular chains of $B\mathrm{Diff}_\partial^{\mathrm{fr}}(D^{2k})$ in high even dimensions, generalizing Goussarov–Habiro graph surgery to a topological realization of Kon. It develops a Brunnian-string-link framework and a hierarchy of vertex-bracket operations (4-, 5-, and higher-valent) realized by embedded Whitehead products and their Jacobi/IHX relations, organized via suspensions/deloopings and thickening to yield compatible chains. It shows that for sufficiently large $k$ the map $\overline{\phi}$ is well-defined and injective on certain excess ranges, with explicit nontrivial cycles arising from excess-2 graphs (e.g., a 5-spoke wheel paired with another graph) that evaluate to Kontsevich invariants via configuration-space integrals. The work culminates in constructing higher-valent Brunnian surgeries that produce nontrivial rational homotopy classes in $\pi_*(B\mathrm{Diff}_\partial^{\mathrm{fr}}(D^{2k}))$, including an explicit 8k−10-dimensional class detected by Kontsevich integrals. Overall, the paper provides a robust topological realization of graph-homology phenomena and yields new rational homotopy classes in high-dimensional diffeomorphism classifying spaces with potential applications to manifold invariants and the study of high-dimensional finite-type schemes.

Abstract

We construct a natural chain map from the Kontsevich graph complex to the rational singular chain complex of $B\mathrm{Diff}_\partial(D^{2k})$ when the dimension $2k$ is sufficiently large, generalizing Goussarov and Habiro's theories of surgery on 3-valent graphs in 3-manifolds. Our construction can be considered as a topological realization of the Kontsevich graph complex. We also give new constructions of elements in the rational homotopy groups of $B\mathrm{Diff}_\partial(D^{2k})$ which are determined by well-known cycles in the graph complex.

Brunnian links and Kontsevich graph complex I

TL;DR

The paper constructs a natural chain map from the Kontsevich graph complex to the rational singular chains of in high even dimensions, generalizing Goussarov–Habiro graph surgery to a topological realization of Kon. It develops a Brunnian-string-link framework and a hierarchy of vertex-bracket operations (4-, 5-, and higher-valent) realized by embedded Whitehead products and their Jacobi/IHX relations, organized via suspensions/deloopings and thickening to yield compatible chains. It shows that for sufficiently large the map is well-defined and injective on certain excess ranges, with explicit nontrivial cycles arising from excess-2 graphs (e.g., a 5-spoke wheel paired with another graph) that evaluate to Kontsevich invariants via configuration-space integrals. The work culminates in constructing higher-valent Brunnian surgeries that produce nontrivial rational homotopy classes in , including an explicit 8k−10-dimensional class detected by Kontsevich integrals. Overall, the paper provides a robust topological realization of graph-homology phenomena and yields new rational homotopy classes in high-dimensional diffeomorphism classifying spaces with potential applications to manifold invariants and the study of high-dimensional finite-type schemes.

Abstract

We construct a natural chain map from the Kontsevich graph complex to the rational singular chain complex of when the dimension is sufficiently large, generalizing Goussarov and Habiro's theories of surgery on 3-valent graphs in 3-manifolds. Our construction can be considered as a topological realization of the Kontsevich graph complex. We also give new constructions of elements in the rational homotopy groups of which are determined by well-known cycles in the graph complex.
Paper Structure (93 sections, 83 theorems, 237 equations, 40 figures)

This paper contains 93 sections, 83 theorems, 237 equations, 40 figures.

Table of Contents

  1. Introduction
  2. Motivation and main results
  3. Basic brackets in the space of string links
  4. String link
  5. Basic bracket operation
  6. General bracket operation
  7. Graph surgery
  8. Organization of the paper
  9. Preliminaries on string links
  10. Some homotopy groups of the spaces of string links
  11. Embeddings with Brunnian null-isotopies
  12. Vertex surgery and framing
  13. Graph surgery
  14. Directed graph cycle
  15. From graphs to chains in $B\mathrm{Diff}_\partial(D^d)$
  16. ...and 78 more sections

Key Result

Theorem 1.1

For $2k\geq 6$, we have outside the "bands" $\bigcup_{n\geq 2}[2kn-4n-1,2kn-1]$ of degrees. The term ${\mathbb Q}$ above is detected by Pontryagin--Weiss classes (We)There is also a major update of the computation of FH by Krannich and Randal-Williams (KrRW) for odd-dimensional disks..

Figures (40)

  • Figure 1: The IHX relation.
  • Figure 2: From a 3-valent graph to the union of genus 3 handlebodies, each of which is identified with the complement of a thickened string link.
  • Figure 3: The graphs $X$ and $Y$.
  • Figure 4: The tree $T_\ell$.
  • Figure 5: The tree $T_\ell(p,q)$, $\ell=p+q$.
  • ...and 35 more figures

Theorems & Definitions (200)

  • Theorem 1.1: KRW
  • Theorem 1: Main Theorem
  • Corollary 2
  • Theorem 3: BWII
  • Corollary 4: BWII
  • Theorem 5: Theorem \ref{['thm:Q-Hurewicz']}
  • Definition 1.3: String link
  • Definition 1.4: Brunnian property
  • Definition 1.5: Brunnian property for family
  • Theorem 1.6: Basic bracket operation
  • ...and 190 more