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On the bridgeless graph complex

Thomas Willwacher

TL;DR

This work analyzes the cohomology of the bridgeless subcomplex $\mathsf{G}_n^{bl}$ of the Kontsevich graph complex, linking it to the ordinary graph complex via the Feynman transform of cyclic operads. It proves a quasi-isomorphism $B_{\mathfrak{k} cyc}^c \mathop{\mathrm{Fe}}(\mathsf{P}) \simeq \mathop{\mathrm{Fe}}(\mathsf{P})^{bl}$ and constructs a convergent spectral sequence $E_2 = H(B_{\mathfrak{k} cyc}^c H(\mathop{\mathrm{Fe}}(\mathsf{P}))) \Rightarrow H(\mathop{\mathrm{Fe}}(\mathsf{P})^{bl})$, enabling the expression of $H(\mathsf{G}_n^{bl})$ in terms of the cohomology of the ordinary Kontsevich graph complex with external legs. The results extend to odd $n$ via 1-shifted cyclic operads and to variants for simple graphs, with a careful analysis showing the projection to simple-graph complexes is a quasi-isomorphism except in a few low-degree cases. A numerical study for the simple bridgeless complex up to loop order $10$ supports the theoretical framework and sheds light on the composition of bridgeless-graph cohomology in terms of tree-assembled components. Overall, the paper reduces the bridgeless cohomology problem to known graph-complex computations with external legs, via a robust operadic and cobar-bar duality approach, and provides practical groundwork for both theoretical and computational exploration.

Abstract

We discuss the cohomology of the bridgeless graph complex, that is, the subcomplex of the Kontsevich graph complex spanned by bridgeless graphs.

On the bridgeless graph complex

TL;DR

This work analyzes the cohomology of the bridgeless subcomplex of the Kontsevich graph complex, linking it to the ordinary graph complex via the Feynman transform of cyclic operads. It proves a quasi-isomorphism and constructs a convergent spectral sequence , enabling the expression of in terms of the cohomology of the ordinary Kontsevich graph complex with external legs. The results extend to odd via 1-shifted cyclic operads and to variants for simple graphs, with a careful analysis showing the projection to simple-graph complexes is a quasi-isomorphism except in a few low-degree cases. A numerical study for the simple bridgeless complex up to loop order supports the theoretical framework and sheds light on the composition of bridgeless-graph cohomology in terms of tree-assembled components. Overall, the paper reduces the bridgeless cohomology problem to known graph-complex computations with external legs, via a robust operadic and cobar-bar duality approach, and provides practical groundwork for both theoretical and computational exploration.

Abstract

We discuss the cohomology of the bridgeless graph complex, that is, the subcomplex of the Kontsevich graph complex spanned by bridgeless graphs.

Paper Structure

This paper contains 8 sections, 6 theorems, 57 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Background: Operads and the Feynman transform
  3. The main results
  4. Proof of Theorem \ref{['thm:main']}
  5. Variant for odd $n$
  6. Variant for simple graphs
  7. Numerical discussion
  8. Proof of Proposition \ref{['prop:multi edge']}

Key Result

Theorem 1.1

Let $\mathsf{P}$ be a 1-shifted cyclic operad. Then there is a quasi-isomorphism of cyclic operads It gives rise to a spectral sequence that converges to the cohomology of $\mathop{\mathrm{Fe}}\nolimits(\mathsf{P})^{bl}$ as long as $\mathsf{P}(r)$ is finite dimensional for each $r$ and zero for $r<3$.

Figures (2)

  • Figure 1: Cohomology dimensions of the bridgeless simple graph complex $\mathsf{GC}_n^{s,bl}$ for $n$ odd (top) and $n$ even (bottom). The rows correspond to the loop orders, the columns to the numbers of vertices. Entries "-" are zero because the complex is zero in those degrees.
  • Figure 2: A copy of the second table of Figure \ref{['fig:cohom']}, but with the different cohomology classes associated to trees of the cyclic bar construction as explained in the text.

Theorems & Definitions (6)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 3.3
  • Corollary 3.4
  • Proposition 3.6
  • Theorem 3.7