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Configuration Spaces of Manifolds with Boundary

Ricardo Campos, Najib Idrissi, Pascal Lambrechts, Thomas Willwacher

Abstract

We study ordered configuration spaces of compact manifolds with boundary. We show that for a large class of such manifolds, the real homotopy type of the configuration spaces only depends on the real homotopy type of the pair consisting of the manifold and its boundary. We moreover describe explicit real models of these configuration spaces using three different approaches. We do this by adapting previous constructions for configuration spaces of closed manifolds which relied on Kontsevich's proof of the formality of the little disks operads. We also prove that our models are compatible with the richer structure of configuration spaces, respectively a module over the Swiss-Cheese operad, a module over the associative algebra of configurations in a collar around the boundary of the manifold, and a module over the little disks operad.

Configuration Spaces of Manifolds with Boundary

Abstract

We study ordered configuration spaces of compact manifolds with boundary. We show that for a large class of such manifolds, the real homotopy type of the configuration spaces only depends on the real homotopy type of the pair consisting of the manifold and its boundary. We moreover describe explicit real models of these configuration spaces using three different approaches. We do this by adapting previous constructions for configuration spaces of closed manifolds which relied on Kontsevich's proof of the formality of the little disks operads. We also prove that our models are compatible with the richer structure of configuration spaces, respectively a module over the Swiss-Cheese operad, a module over the associative algebra of configurations in a collar around the boundary of the manifold, and a module over the little disks operad.

Paper Structure

This paper contains 65 sections, 80 theorems, 296 equations, 14 figures, 3 tables.

Table of Contents

  1. Background and recollections
  2. Colored (co)operads and (co)modules
  3. Operads over spaces and right multimodules
  4. A note on the cohomology of compact orientable manifolds with boundary
  5. Poincaré--Lefschetz duality models
  6. Motivation
  7. Definition and existence
  8. Diagonal classes
  9. Configuration spaces, their compactifications, and algebraic structures
  10. Semi-algebraic realization
  11. (Axelrod--Singer--)Fulton--MacPherson operad FM_n
  12. Fiberwise little n-disks operad
  13. Compactified configuration space for closed manifolds
  14. Swiss Cheese operad and fiberwise Swiss Cheese operad
  15. Compactified configuration space for manifolds with boundary
  16. ...and 50 more sections

Key Result

Theorem 1

Let $M$ be an oriented compact manifold with boundary $\partial M\neq \emptyset$. Then there is zigzag of quasi-isomorphisms relating the pairs compatible with all structures, i.e., with the dg commutative algebra structure and the operadic action of the second member of the pairs on the first.

Figures (14)

  • Figure 3.1: Strata for $\mathsf{FM}_n(r)$
  • Figure 3.2: Strata for $\mathsf{FM}_M(r)$
  • Figure 3.3: A tree indexing a stratum of $\mathsf{SFM}_M(1,6)$ and a point in that stratum
  • Figure 3.4: Adjacency of boundary strata in $\mathsf{aFM}_N(2)$
  • Figure 3.5: Adjacency of boundary strata in $\mathsf{mFM}_M(2)$
  • ...and 9 more figures

Theorems & Definitions (188)

  • Theorem 1: See Theorem \ref{['thm.main-qiso']}
  • Corollary 2: Corollary \ref{['cor.inv-htp1']}
  • Theorem 3: See Section \ref{['sec:secondmodel']}
  • Corollary 4
  • Theorem 5: See Theorems \ref{['thm.model-ga']}--\ref{['thm.ext-formal']}
  • Remark 6
  • Definition 1.1
  • Remark 1.2
  • Example 1.3
  • Lemma 1.4
  • ...and 178 more