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Configuration spaces of clusters as $E_d$-algebras

Florian Kranhold

Abstract

It is a classical result that configuration spaces of labelled particles in $\mathbb{R}^d$ are free $E_d$-algebras and that their $d$-fold bar construction is equivalent to the $d$-fold suspension of the labelling space. In this paper, we study a variation of these spaces, namely configuration spaces of labelled clusters of particles. These configuration spaces are again $E_d$-algebras, and we give geometric models for their iterated bar construction in two different ways: one establishes a description of these configuration spaces of clusters as cellular $E_1$-algebras, and the other one uses an additional verticality constraint. In the last section, we apply these results in order to calculate the stable homology of certain vertical configuration spaces.

Configuration spaces of clusters as $E_d$-algebras

Abstract

It is a classical result that configuration spaces of labelled particles in are free -algebras and that their -fold bar construction is equivalent to the -fold suspension of the labelling space. In this paper, we study a variation of these spaces, namely configuration spaces of labelled clusters of particles. These configuration spaces are again -algebras, and we give geometric models for their iterated bar construction in two different ways: one establishes a description of these configuration spaces of clusters as cellular -algebras, and the other one uses an additional verticality constraint. In the last section, we apply these results in order to calculate the stable homology of certain vertical configuration spaces.

Paper Structure

This paper contains 9 sections, 7 theorems, 28 equations, 6 figures.

Table of Contents

  1. Introduction and overview
  2. Related work
  3. Outlook
  4. Acknowledgements
  5. Funding
  6. Basic constructions
  7. Cellular decompositions of clustered configuration spaces
  8. Iterated bar constructions of vertical configuration spaces
  9. Stable homology of vertical configuration spaces

Key Result

Theorem 3.4

Let $\bm{X}=(X_k)_{k\ge 1}$ be a sequence of well-based spaces (with arbitrary based $\mathfrak{S}_k$-actions on $X_k$). Then we have a weak equivalence, abbreviating $\bm{X}^{\wedge K}\coloneqq X_{k_1}\wedge \dotsb\wedge X_{k_r}$,

Figures (6)

  • Figure 1: Several configuration spaces of three $2$-clusters inside $\mathbb{R}^2$ and $\mathbb{R}^3$.
  • Figure 2: An instance of $\lambda_3\colon \mathscr{C}_2(3)\times C(\mathbb{R}^2;\uline{{\mathbb{S}}}^0)^3 \to C(\mathbb{R}^2;\uline{{\mathbb{S}}}^0)$
  • Figure 3: An instance of $\chi\colon C(\mathbb{R};\uline{{\mathbb{S}}}^0)\to \Xi$
  • Figure 4: An instance of the map $C"_{p,pxq}(\bm{X}) \to D_{p-1,pxq}(\bm{X})$
  • Figure 5: The left configuration is smaller than the right one since it is a restriction of the latter.
  • ...and 1 more figures

Theorems & Definitions (32)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.5: Wreath products
  • Definition 2.7
  • Remark 2.8
  • Definition 3.1
  • Example 3.3
  • Theorem 3.4
  • Example 3.5
  • ...and 22 more