In this paper, we introduce and study representation homology of topological spaces, which is a natural homological extension of representation varieties of fundamental groups. We give an elementary construction of representation homology parallel to the Loday–Pirashvili construction of higher Hochschild homology; in fact, we establish a direct geometric relation between the two theories by proving that the representation homology of the suspension of a (pointed connected) space is isomorphic to its higher Hochschild homology. We also construct some natural maps and spectral sequences relating representation homology to other homology theories associated with spaces (such as Pontryagin algebras, ${{\mathbb{S}}}^1$-equivariant homology of the free loop space, and stable homology of automorphism groups of f.g. free groups). We compute representation homology explicitly (in terms of known invariants) in a number of interesting cases, including spheres, suspensions, complex projective spaces, Riemann surfaces, and some 3-dimensional manifolds, such as link complements in ${\mathbb{R}}^3$ and the lens spaces $ L(p,q) $. In the case of link complements, we identify the representation homology in terms of ordinary Hochschild homology, which gives a new algebraic invariant of links in ${\mathbb{R}}^3$.
We generalize Keller's construction [38] of deformed n-Calabi-Yau completions to the relative contexts. This gives a universal construction which extends any given DG functor F : A → B to a DG functor F : Ã → B, together with a family of deformations of F parametrized by relative negative cyclic homology classes [η] ∈ HC − n−2 (B, A). We show that, under a finiteness condition, these extensions have canonical relative n-Calabi-Yau structures in the sense of [8].This is applied to give a construction which associates a DG category A (N, M ; Φ) to a pair (N, M ) consisting of a manifold N and an embedded submanifold M of codimension ≥ 2, together with a trivialization Φ of the unit normal bundle of M in N . In the case when (N, M ) is the pair consisting of a set of n-points in the interior of the 2-dimensional disk, A (N, M ; Φ) is the multiplicative preprojective algebra [16] with non-central parameters. In the case when (N, M ) is the pair consisting of a link L in R 3 , A (N, M ; Φ) is the link DG category [5] that extends the Lengendrian DG algebra [45,46,24] of the unit conormal bundle ST * L (R 3 ) ⊂ ST * (R 3 ). We show that, when M ⊂ N has codimension 2, then the category of finite dimensional modules over the 0-th homology H 0 ( A (N, M ; Φ) ) of this DG category is equivalent to the category of perverse sheaves on N with singularities at most along M .
No abstract
Let G be a reductive affine algebraic group defined over a field k of characteristic zero. In this paper, we study the cotangent complex of the derived G-representation scheme DRep G (X) of a pointed connected topological space X. We use an (algebraic version of) unstable Adams spectral sequence relating the cotangent homology of DRep G (X) to the representation homology HR * (X, G) := π * O[DRep G (X)] to prove some vanishing theorems for groups and geometrically interesting spaces. Our examples include virtually free groups, Riemann surfaces, link complements in R 3 and generalized lens spaces. In particular, for any f.g. virtually free group Γ, we show that HR i (BΓ, G) = 0 for all i > 0. For a closed Riemann surface Σg of genus g ≥ 1, we have HR i (Σg, G) = 0 for all i > dim G. The sharp vanishing bounds for Σg depend actually on the genus: we conjecture that if g = 1, then HR i (Σg , G) = 0 for i > rank G, and if g ≥ 2, then HR i (Σg, G) = 0 for i > dim Z(G) , where Z(G) is the center of G. We prove these bounds locally on the smooth locus of the representation scheme Rep G [π 1 (Σg )] in the case of complex connected reductive groups. One important consequence of our results is the existence of a well-defined K-theoretic virtual fundamental class for DRep G (X) in the sense of Ciocan-Fontanine and Kapranov [12]. We give a new "Tor formula" for this class in terms of functor homology.
We develop the technique of weight truncation in the context of wall-crossings in birational cobordisms, parallel to that in [7,4]. More precisely, for each such wall-crossing, we embed the bounded above derived category of coherent sheaves of the semistable part as a semi-orthogonal summand of that of the stack in question. Our construction does not require any smoothness assumptions, and exhibits a strong symmetry across the two sides of the wall-crossing. As an application, we show that for wallcrossings satisfying suitable regularity conditions, a certain duality of local cohomology complexes implies the existence of a fully faithful functor/equivalence between the derived categories under wall-crossings.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.