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Koszul self duality of manifolds

Connor Malin

Abstract

We show that Koszul duality for operads in $(\mathrm{Top},\times)$ can be expressed via generalized Thom complexes. As an application, we prove the Koszul self duality of the little disk modules $E_M$. We discuss implications for factorization homology, embedding calculus, and confirm an old conjecture of Ching on the relation of Goodwillie calculus to manifold calculus.

Koszul self duality of manifolds

Abstract

We show that Koszul duality for operads in can be expressed via generalized Thom complexes. As an application, we prove the Koszul self duality of the little disk modules . We discuss implications for factorization homology, embedding calculus, and confirm an old conjecture of Ching on the relation of Goodwillie calculus to manifold calculus.
Paper Structure (9 sections, 40 theorems, 173 equations)

This paper contains 9 sections, 40 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Operads and right modules of interest
  3. Ching's conjecture
  4. Parametrized spectra
  5. Verdier Duality
  6. Koszul-Verdier duality of operads and right modules
  7. Self duality for submanifolds of $\mathbb{R}^n$
  8. Weiss cosheaves in the category $\mathrm{RMod}_{O}$
  9. Self duality of $E_M$, Poincaré-Koszul duality, and embedding calculus

Key Result

Proposition 1.1

To an operad $O$ and right module pair $(R,A)$ in $(\mathrm{Top},\times)$ we can associate an operad $\xi_O$ and right module $\xi_{(R,A)}$ in $(\mathrm{ParSp},\bar{\wedge})$ for which there are compatible equivalences

Theorems & Definitions (153)

  • Proposition 1.1: Propositions \ref{['prp:koszulverdieroperad']}, \ref{['prp:koszulverdiermodule']}: Koszul-Verdier duality
  • Theorem 1.2: Theorem \ref{['thm:selfduality']}: Koszul self duality of $E_M$
  • Corollary 1.3: Corollary \ref{['cor:chingconjecture']}: Ching's conjecture
  • Theorem 1.4: Theorem \ref{['thm:pontry']}: Pontryagin-Thom equivalence
  • Theorem 1.5: Theorem \ref{['thm:Poincarékoszul']}: Poincaré-Koszul duality for left $\Sigma^\infty_+ E_n$-modules
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 143 more