Homological stability for topological chiral homology of completions

Abstract

Abstract. By proving that several new complexes of embedded disks are highly connected, we obtain several new homological stability results. Our main result is homological stability for topological chiral homology on an open manifold with coefficients in certain partial framed En-algebras. Using this, we prove a special case of a conjecture of Vakil and Wood on homological stability for complements of closures of particular strata in the symmetric powers of an open manifold and we prove that the bounded symmetric powers of closed manifolds satisfy homological stability rationally.

Discriminants in the Grothendieck ring

“…(As in the case of w 1 j (X), there do not exist obvious maps among the elements of these sequences of configuration spaces for closed X; many topological stabilization results rely on such a map.) Since the authors publicly made this conjecture in the first draft of this paper, Kupers and Miller have proven Conjecture F in the case that λ = m for an integer m, for X a "reasonable" manifold [KM2]. This can be done using the topological methods of [Mc2], but Kupers and Miller can also show homological stability with a specific range, and moreover their result holds with Z-coefficients so long as the manifold is not closed.…”

“…Since we made these conjectures publicly in the first draft of our paper, they have motivated a significant amount of research by several authors into the structure that our conjectures highlight, e.g [Ch2,Cor. 3] and [KM1,KM2,Tom1,Tom2]. Some of the conjectures have been proven in special cases, some have been proven entirely, and some have been disproven.…”

Homological stability for spaces of embedded surfaces

“…He considers the space of all embedded submanifolds of a chosen diffeomorphism type in a background manifold satisfying certain hypotheses, and stabilises by repeatedly adding disjoint copies of a chosen submanifold near the boundary of the background manifold. The case of 1-dimensional manifolds in R 3 is not covered by Palmer's result, but is nonetheless true [Kup13].…”

$$E_n$$ E n -cell attachments and a local-to-global principle for homological stability

“…For any oriented connected non-compact n-dimensional manifold A, we have that t : This proposition has implications for completions (in the category of N 0 -charged algebras) of [d]-charged algebras without making any assumptions on their homology. Such a result was proven in Theorem 1.1 of [7] but with a worse range. The techniques of [7] are completely different than those used here and follow the traditional approach to proving homological stability introduced by Quillen.…”

“…, x k ) ∈ (R n ) k | x i = x j if i = j}/S k with S k the symmetric group on k letters. Other examples of framed E n -algebras with homological stability include symmetric powers of R n [3], bounded symmetric powers of R n [4] [5], various decorated configuration spaces [6], completions of certain partial E n -algebras [7], some spaces of branched covers [8], classifying spaces of groups of diffeomorphisms fixing a disk [9] [10] [11], moduli spaces of manifolds embedded in R n [12], moduli spaces of instantons [13] and spaces of rational or holomorphic functions [2] [14] [15] [16]. In all of these examples, the map eventually inducing isomorphisms on homology is constructed using the framed E n -algebra structure.…”