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Heisenberg homology of ribbon graphs

Christian Blanchet

TL;DR

The paper generalizes Heisenberg homology of configuration spaces from surfaces with a single boundary to oriented surfaces $\Sigma_{g,m}$ with multiple boundary components, by using a regular cover associated to the Heisenberg group $\mathcal{H}(\Sigma_{g,m})$ and twisted local coefficients. It then establishes a compression trick showing that graph configuration BM homology matches the BM homology of the thickened surface, while highlighting that the induced local system depends on the chosen ribbon thickening. A finite BM cellular complex for graph configurations is developed, enabling explicit computation of $H_*^{BM}(\mathcal{C}_n(\Gamma);W)$ and its relative version, and connecting to surface data via thickened relative ribbon graphs. The action of the Mapping Class Group on Heisenberg data is formalized, including a natural twisted action on homology with coefficient twists, and representations of the MCG are produced from the regular Heisenberg action. The paper also provides an explicit computational framework (illustrated by a genus $1$ with $1$ boundary example) using Fox calculus to compute twist matrices and verify braid relations, paving the way for practical calculations of monodromy and related invariants in curve singularity contexts.

Abstract

We review Heisenberg homology of configurations in once bounded surfaces and extend the construction to the regular thickening of a finite graph with ribbon structure.

Heisenberg homology of ribbon graphs

TL;DR

The paper generalizes Heisenberg homology of configuration spaces from surfaces with a single boundary to oriented surfaces with multiple boundary components, by using a regular cover associated to the Heisenberg group and twisted local coefficients. It then establishes a compression trick showing that graph configuration BM homology matches the BM homology of the thickened surface, while highlighting that the induced local system depends on the chosen ribbon thickening. A finite BM cellular complex for graph configurations is developed, enabling explicit computation of and its relative version, and connecting to surface data via thickened relative ribbon graphs. The action of the Mapping Class Group on Heisenberg data is formalized, including a natural twisted action on homology with coefficient twists, and representations of the MCG are produced from the regular Heisenberg action. The paper also provides an explicit computational framework (illustrated by a genus with boundary example) using Fox calculus to compute twist matrices and verify braid relations, paving the way for practical calculations of monodromy and related invariants in curve singularity contexts.

Abstract

We review Heisenberg homology of configurations in once bounded surfaces and extend the construction to the regular thickening of a finite graph with ribbon structure.
Paper Structure (8 sections, 9 theorems, 40 equations, 3 figures)

This paper contains 8 sections, 9 theorems, 40 equations, 3 figures.

Table of Contents

  1. Heisenberg homology of surface configurations
  2. Ribbon graphs
  3. A complex for Borel-Moore homology of graph configurations
  4. Action of mapping classes
  5. Action on the Heisenberg group
  6. Twisted action of the Mapping Class Group
  7. MCG representations from the regular action on Heisenberg group
  8. About computation

Key Result

Proposition 1

For each $g\geqslant 0$, $m\geqslant 1$ and $n\geqslant 2$, the quotient of the braid group $\mathbb{B}_n(\Sigma_{g,m})$ by the subgroup $[\sigma_1,\mathbb{B}_n(\Sigma_{g,m})]^N$ normally generated by the commutators $[\sigma_1,x]$, $x\in \mathbb{B}_n(\Sigma_{g,m})$, is isomorphic to the Heisenberg is represented by the surjective homomorphism sending each $\sigma_i$ to $u=(1,0)$, $\alpha_r$ to

Figures (3)

  • Figure 1: Model for $\Sigma_{g,m}$.
  • Figure 2: Surfaces $\Sigma_\Gamma$ and $\Sigma_{(\Gamma,A)}$ for a relative graph $(\Gamma,A)$, where $A$ is the horizontal interval.
  • Figure 3: Surface $\Sigma_{1,1}$ with tether from the base $2$ points configuration to $\alpha$ and $\beta$.

Theorems & Definitions (11)

  • Proposition 1
  • Theorem 2
  • Definition 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Proposition 7
  • Theorem 8
  • Theorem 9
  • Proposition 10
  • ...and 1 more