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On coshuffle comultiplication on configuration spaces

Byung Hee An

TL;DR

The paper defines a coshuffle comultiplication $\Sha^*$ on the singular chain complex of configuration spaces, turning them into differential graded coalgebras ($\mathsf{DGCoAlg}$) and showing compatibility with external product. It then specializes to graphs via the Świątkowski complexes, proving the coshuffle structure transfers and giving an explicit formula for $\Sha^*_S$ on generators. For graphs of topological circumference at most 1 (bunches of grapes), the authors prove formality of the configuration-space coalgebras and provide a complete description of primitive homology classes, expressed in terms of edge data, star-cycle, and loop generators. The results illuminate the interplay between combinatorial graph data and coalgebraic structures on configuration spaces, with concrete consequences for graph braid groups and configuration-space topology.

Abstract

We introduce a coshuffle comultiplication on the singular chain complex of configuration spaces, and we show that this structure endows the configuration space with the structure of a differential graded coalgebra (DGCoAlg). We then prove that the coshuffle comultiplication is compatible with the external product through a natural commutation relation. As an application, we investigate configuration spaces of graphs and the associated graph braid groups. In particular, for graphs of topological circumference at most 1, we prove that the singular chain complex of the configuration space is formal as a DGCoAlg. Moreover, we obtain a complete classification of the primitivity in the homology of configuration spaces of such graphs.

On coshuffle comultiplication on configuration spaces

TL;DR

The paper defines a coshuffle comultiplication on the singular chain complex of configuration spaces, turning them into differential graded coalgebras () and showing compatibility with external product. It then specializes to graphs via the Świątkowski complexes, proving the coshuffle structure transfers and giving an explicit formula for on generators. For graphs of topological circumference at most 1 (bunches of grapes), the authors prove formality of the configuration-space coalgebras and provide a complete description of primitive homology classes, expressed in terms of edge data, star-cycle, and loop generators. The results illuminate the interplay between combinatorial graph data and coalgebraic structures on configuration spaces, with concrete consequences for graph braid groups and configuration-space topology.

Abstract

We introduce a coshuffle comultiplication on the singular chain complex of configuration spaces, and we show that this structure endows the configuration space with the structure of a differential graded coalgebra (DGCoAlg). We then prove that the coshuffle comultiplication is compatible with the external product through a natural commutation relation. As an application, we investigate configuration spaces of graphs and the associated graph braid groups. In particular, for graphs of topological circumference at most 1, we prove that the singular chain complex of the configuration space is formal as a DGCoAlg. Moreover, we obtain a complete classification of the primitivity in the homology of configuration spaces of such graphs.

Paper Structure

This paper contains 15 sections, 30 theorems, 113 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The results of the paper
  3. Outline of the paper
  4. Acknowledgments
  5. (Co)Shuffle (Co)multiplications
  6. (Co)Shuffle (co)multiplication on configuration spaces
  7. External product and primitivity
  8. Comultiplication on configuration spaces of graphs
  9. The Świątkowski complexes for graphs
  10. Comultiplication $\Sha^*_S$ on $S_\bullet(\mathsf{\Gamma})$
  11. Comultiplication ${\Sha}^*$ on $R[E]$
  12. Applications to bunches of grapes
  13. Bunches of grapes
  14. The formality
  15. The primitivity

Key Result

Theorem 1.1

Let $e:C_\bullet(BX)\to C_0(B_0X)\cong R$ be the projection map onto the summand of degree $0$ and weight $0$, and let $u:R\cong C_0(B_0X)\to C_\bullet(BX)$ be the map induced by the inclusion. Then $(C_\bullet(BX),\Sha^*,e,u)$ is a $\mathsf{DGCoAlg}$.

Figures (4)

  • Figure 1: Example of a $\Sha^*_{(1,0),\mathbf{k}}$.
  • Figure 2: A bunch of grapes (left) and an elementary bunch of grapes (right).
  • Figure 3: The decomposition of $\mathsf{\Gamma}$ along the stem.
  • Figure 4: Labeling convention of edges in $\Gamma$ and its decomposition.

Theorems & Definitions (69)

  • Theorem 1.1: \ref{['theorem:coshuffle']}
  • Theorem 1.2: \ref{['theorem:shuffle product on cohomology']}
  • Theorem 1.3: \ref{['theorem:formality']}
  • Theorem 1.4: \ref{['theorem:primitivity']}
  • Definition 2.1: Intertwining configuration spaces
  • Example 2.2
  • Definition 2.3: Coshuffle maps
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • ...and 59 more