We generalize the nonabelian Poincaré 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 "bringing 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, M A, is homology equivalent to M ap c (M, B n A), the space of compactly supported maps to the n-fold classifying space of A. Topological chiral homology via the twosided bar constructionIn this subsection, we will describe a definition of topological chiral homology derived from [And10] employing the monadic two-sided bar construction. The monadic two-sided bar construction was introduced in [May72] to prove the recognition principle, the theorem that all connected algebras over the little n-disks operad are homotopy equivalent to n-fold loop spaces. May's

Using the language of twisted skew-commutative algebras, we define secondary representation stability, a stability pattern in the unstable homology of spaces that are representation stable in the sense of Church-Ellenberg-Farb [CEF15]. We show that the rational homology of configuration spaces of ordered points in noncompact manifolds satisfies secondary representation stability. While representation stability for the homology of configuration spaces involves stabilizing by introducing a point "near infinity," secondary representation stability involves stabilizing by introducing a pair of orbiting points -an operation that relates homology groups in different homological degrees. This result can be thought of as a representation-theoretic analogue of secondary homological stability in the sense of Galatius-Kupers-Randal-Williams [GKRWa, GKRWb]. In the course of the proof we establish some additional results: we give a new characterization of the homology of the complex of injective words, and we give a new proof of integral representation stability for configuration spaces of noncompact manifolds, extending previous results to nonorientable manifolds.

We define bounded generation for En-algebras in chain complexes and prove that for n ≥ 2 this property is equivalent to homological stability. Using this we prove a local-to-global principle for homological stability, which says that if an En-algebra A has homological stability (or equivalently the topological chiral homology R n A has homology stability), then so has the topological chiral homology M A of any connected non-compact manifold M . Using scanning, we reformulate the local-to-global homological stability principle so that it applies to compact manifolds. We also give several applications of our results.Here k M X denotes the charge k component of M X, B T M X is a bundle over M with fiber given by the n-fold delooping B n X of X, and Γ c k (−) denotes the space of compactly

The purpose of this note is to clarify some details in McDuff and Segal's proof of the group-completion theorem in [MS75] and generalize this and the homology fibration criterion of [McD75] to homology with twisted coefficients. This will be used in [MP] to identify the limiting homology of "oriented" configuration spaces, which doubly cover the classical configuration spaces of distinct unordered points in a manifold.

In [Chu12], Church used representation stability to prove that the space of configurations of distinct unordered points in a closed manifold exhibit rational homological stability. A second proof was also given by Randal-Williams in [RW13] using transfer maps. We give a third proof of this fact using localization and rational homotopy theory. This gives new insight into the role that the rationals play in homological stability. Our methods also yield new information about stability for torsion in the homology of configuration spaces of points in a closed manifold. ScanningAlthough McDuff did not address the question of homological stability for configuration spaces of points in a closed manifold, she did prove the following theorem (Theorem 1.1 of [McD75]).