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Symmetric homology is a natural generalization of cyclic homology, in which symmetric groups play the role of cyclic groups. In the case of associative algebras, the symmetric homology theory was introduced by Z. Fiedorowicz (1991) and was further developed in the work of S. Ault (2010). In this paper, we show that, for algebras defined over a field of characteristic 0, the symmetric homology is naturally equivalent to the (one-dimensional) representation homology introduced by the authors in joint work with G. Khachatryan (2013). Using known results on representation homology, we compute symmetric homology explicitly for basic algebras, such as polynomial algebras and universal enveloping algebras of (DG) Lie algebras. As an application, we prove two conjectures of Ault and Fiedorowicz, including their main conjecture (2007) on topological interpretation of symmetric homology of polynomial algebras. [ABSTRACT FROM AUTHOR]

Secondary homological stability is a recently discovered stability pattern for a sequence of spaces exhibiting homological stability and it holds outside the range where the homology stabilizes. We prove secondary homological stability for the homology of the unordered configuration spaces of a connected manifold. The main difficulty is the case that the manifold is closed because there are no obvious maps inducing stability and the homology eventually is periodic instead of stable. We resolve this issue by constructing a new chain-level stabilization map for configuration spaces. [ABSTRACT FROM AUTHOR]

Let A be a noetherian connected graded algebra. We introduce and study homological invariants that are weighted sums of the homological and internal degrees of cochain complexes of graded A-modules, providing weighted versions of Castelnuovo–Mumford regularity, Tor-regularity, Artin–Schelter regularity, and concavity. In some cases an invariant (such as Tor-regularity) that is infinite can be replaced with a weighted invariant that is finite, and several homological invariants of complexes can be expressed as weighted homological regularities. We prove a few weighted homological identities some of which unify different classical homological identities and produce interesting new ones. [ABSTRACT FROM AUTHOR]

We compute the Reidemeister-Turaev torsion of homology cylinders which takes values in the K_1-group of the I-adic completion of the group ring \mathbb {Q}\pi _1\Sigma _{g,1}, and prove that its reduction to \widehat {\mathbb {Q}\pi _1\Sigma _{g,1}}/\hat {I}^{d+1} is a finite-type invariant of degree d. We also show that the 1-loop part of the LMO homomorphism and the Enomoto-Satoh trace can be recovered from the leading term of our torsion. [ABSTRACT FROM AUTHOR]

Quandle homology theory has been developed and cocycles have been used to define invariants of oriented classical or surface links. We introduce a shifting chain map \sigma on each quandle chain complex that lowers the dimensions by one. By using its pull-back \sigma ^\sharp, each 2-cocycle \phi gives us the 3-cocycle \sigma ^\sharp \phi. For oriented classical links in the 3-space, we explore relation between their quandle 2-cocycle invariants associated with \phi and their shadow 3-cocycle invariants associated with \sigma ^\sharp \phi. For oriented surface links in the 4-space, we explore how powerful their quandle 3-cocycle invariants associated with \sigma ^\sharp \phi are. Algebraic behavior of the shifting maps for low-dimensional (co)homology groups is also discussed. [ABSTRACT FROM AUTHOR]

We provide infinitely many rational homology 3-spheres with weight-one fundamental groups which do not arise from Dehn surgery on knots in S3. In contrast with previously known examples, our proofs do not require any gauge theory or Floer homology. Instead, we make use of the SU(2) character variety of the fundamental group, which for these manifolds is particularly simple: they are all SU(2)-cyclic, meaning that every SU(2) representation has cyclic image. Our analysis relies essentially on Gordon-Luecke's classification of half-integral toroidal surgeries on hyperbolic knots, and other classical 3-manifold topology. [ABSTRACT FROM AUTHOR]

We describe new Chow-weight (co)homology theories on the category DM^{\mathrm {eff}}_{gm}(k,R) of effective geometric Voevodsky motives (R is the coefficient ring). These theories are interesting "modifications" of motivic homology; Chow-weight homology detects whether a motive M\in ObjDM^{\mathrm {eff}}_{gm}(k,R) is r-effective (i.e., belongs to the rth Tate twist DM^{\mathrm {eff}}_{gm}(k,R)(r) of effective motives), bounds the weights of M (in the sense of the Chow weight structure defined by the first author), and detects the effectivity of "the lower weight pieces" of M. Moreover, we calculate the connectivity of M (in the sense of Voevodsky's homotopy t-structure, i.e., we study motivic homology) and prove that the exponents of the higher motivic homology groups (of an "integral" motive) are finite whenever these groups are torsion. We apply the latter statement to the study of higher Chow groups of arbitrary varieties. These motivic properties of M have plenty of applications. They are closely related to the (co)homology of M; in particular, if the Chow groups of a variety X vanish up to dimension r-1 then the highest Deligne weight factors of the (singular or étale) cohomology of X with compact support are r-effective. Our results yield vast generalizations of the so-called "decomposition of the diagonal" theorems, and we re-prove and extend some of earlier statements of this sort. [ABSTRACT FROM AUTHOR]

In the spirit of Hjorth's turbulence theory, we introduce "unbalancedness": a new dynamical obstruction to classifying orbit equivalence relations by actions of Polish groups which admit a two-sided invariant metric (TSI). Since abelian groups are TSI, unbalancedness can be used for identifying which classification problems cannot be solved by classical homology and cohomology theories. In terms of applications, we show that Morita equivalence of continuous-trace C*-algebras, as well as isomorphism of Hermitian line bundles, are not classifiable by actions of TSI groups. In the process, we show that the Wreath product of any two non-compact subgroups of S∞ admits an action whose orbit equivalence relation is generically ergodic against any action of a TSI group and we deduce that there is an orbit equivalence relation of a CLI group which is not classifiable by actions of TSI groups. [ABSTRACT FROM AUTHOR]