Nonabelian Poincare duality after stabilizing

TL;DR

This paper extends nonabelian Poincare duality to non-grouplike E_n-algebras, introducing a stabilization process that equates topological chiral homology with a space of maps, broadening the duality's applicability.

Contribution

It generalizes existing duality theorems to a wider class of E_n-algebras by defining a stabilization procedure and proving an equivalence with a mapping space.

Findings

Stabilization makes non-grouplike E_n-algebras compatible with duality.

Topological chiral homology is homology equivalent to a space of compactly supported maps.

The results apply to open connected parallelizable n-manifolds.

Abstract

We generalize the nonabelian Poincare duality theorems of Salvatore in [Sal01] and Lurie in [Lur09] to the case of not necessarily grouplike E_n-algebras (in the category of spaces). We define a stabilization procedure based on McDuff's "brining points in from infinity" maps from [McD75]. For open connected parallelizable n-manifolds, we prove that, after stabilizing, the topological chiral homology of M with coefficients in an E_n-algebra A, is homology equivalent to Map^c(M,B^n A), the space of compactly supported maps to the n-fold classifying space of A. The two models of topological chiral homology used in this paper are Andrade's model from [And10] and Salvatore's from [Sal01].