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Let \pi \colon Y\to X denote the canonical resolution of the two dimensional Kleinian singularity X of type ADE. In the present paper, we establish isomorphisms between the cohomological and K-theoretical Hall algebras of \omega-semistable properly supported sheaves on Y with fixed slope \mu and \zeta-semistable finite-dimensional representations of the preprojective algebra of affine type ADE of slope zero respectively, under some conditions on \zeta depending on the polarization \omega and \mu. These isomorphisms are induced by the derived McKay correspondence. In addition, they are interpreted as decategorified versions of a monoidal equivalence between the corresponding categorified Hall algebras. In the type A case, we provide a finer description of the cohomological, K-theoretical and categorified Hall algebra of \omega-semistable properly supported sheaves on Y with fixed slope \mu: for example, in the cohomological case, the algebra can be given in terms of Yangians of finite type ADE Dynkin diagrams. [ABSTRACT FROM AUTHOR]

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]

In this paper, we build up a scaled homology theory, lc-homology, for metric spaces such that every metric space can be visually regarded as "locally contractible" with this newly-built homology. We check that lc-homology satisfies all Eilenberg-Steenrod axioms except the exactness axiom whereas its corresponding lc-cohomology satisfies exactness axiom for cohomology. This homology can relax the smooth manifold restrictions on the compact metric space such that the entropy conjecture will hold for the first lc-homology group. [ABSTRACT FROM AUTHOR]

A well-known conjecture of Rasmussen states that for any knot K in S^{3}, the rank of the reduced Khovanov homology of K is greater than or equal to the rank of the reduced knot Floer homology of K. This rank inequality is supposed to arise as the result of a spectral sequence from Khovanov homology to knot Floer homology. Using an oriented cube of resolutions construction for a homology theory related to knot Floer homology, we prove this conjecture. [ABSTRACT FROM AUTHOR]

It is well known that every homology functor on the stable homotopy category is representable, so of the form E*t(X) = Π*(E Λ X) for some spectrum E. However, Christensen, Keller, and Neeman (2001) have exhibited simple triangulated categories, such as the derived category of k[x, y] for sufficiently large fields k, for which not every homology functor is representable. In this paper, we show that this failure of Brown representability does not happen on the model category level. That is, we show that a homology theory is representable if and only if it lifts to a well-behaved functor on the model category level. We also show that, for a reasonable model category M, every functor that has the same formal properties as a functor of the form X → X ⊗ E for some cofibrant E is naturally weakly equivalent to a functor of that form. This is closely related to the Eilenberg-Watts theorem in algebra, which proves that every functor with the same formal properties as the tensor product with a fixed object is isomorphic to such a functor. [ABSTRACT FROM AUTHOR]

In this brief note, we investigate the \mathbb {CP}^2-genus of knots, i.e., the least genus of a smooth, compact, orientable surface in \mathbb {CP}^2\smallsetminus \mathring {B^4} bounded by a knot in S^3. We show that this quantity is unbounded, unlike its topological counterpart. We also investigate the \mathbb {CP}^2-genus of torus knots. We apply these results to improve the minimal genus bound for some homology classes in \mathbb {CP}^2\# \mathbb {CP} ^2. [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 k be an algebraically closed field and A a (left and right) Noetherian associative k-algebra. Assume further that A is either positively graded or semiperfect (this includes the class of finite dimensional k-algebras and k-algebras that are finitely generated modules over a Noetherian central Henselian ring). Let e be a primitive idempotent of A, which we assume is of degree 0 if A is positively graded. We consider the idempotent subalgebra Γ = (1-e)A(1-e) and Se the simple right A-module Se = eA/eradA, where radA is the Jacobson radical of A or the graded Jacobson radical of A if A is positively graded. In this paper, we relate the homological dimensions of A and Γ, using the homological properties of Se. First, if Se has no self-extensions of any degree, then the global dimension of A is finite if and only if that of Γ is. On the other hand, if the global dimensions of both A and Γ are finite, then Se cannot have self-extensions of degree greater than one, provided A/radA is finite dimensional. [ABSTRACT FROM AUTHOR]

Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently observed in nature. In this paper a theoretical framework for the algorithmic computation of an arbitrarily good approximation of the persistent homology is developed. We study the filtrations generated by sub-level sets of a function f : X → R, where X is a CW-complex. In the special case X = [0, 1]N, N ∈ N, we discuss implementation of the proposed algorithms. We also investigate a priori and a posteriori bounds of the approximation error introduced by our method. [ABSTRACT FROM AUTHOR]

In this paper, certain connections between complex symmetric operators and anti-automorphisms of singly generated C*-algebras are established. This provides a C*-algebra approach to the norm closure problem for complex symmetric operators. For T ∈ B(H) satisfying C*(T)∩K(H) = {0}, we give several characterizations for T to be a norm limit of complex symmetric operators. As applications, we give concrete characterizations for weighted shifts with nonzero weights to be norm limits of complex symmetric operators. In particular, we prove a conjecture of Garcia and Poore. On the other hand, it is proved that an essentially normal operator is a norm limit of complex symmetric operators if and only if it is complex symmetric. We obtain a canonical decomposition for essentially normal operators which are complex symmetric. [ABSTRACT FROM AUTHOR]

We define the categorical cohomology of a k-graph Λ and show that the first three terms in this cohomology are isomorphic to the corresponding terms in the cohomology defined in our previous paper. This leads to an alternative characterisation of the twisted k-graph C*-algebras introduced there. We prove a gauge-invariant uniqueness theorem and use it to show that every twisted k-graph C*-algebra is isomorphic to a twisted groupoid C*-algebra. We deduce criteria for simplicity, prove a Cuntz-Krieger uniqueness theorem and establish that all twisted k-graph C*-algebras are nuclear and belong to the bootstrap class. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the polarization of Koszul cycles and use it to study the depth function of powers of edge ideals of whisker graphs. [ABSTRACT FROM AUTHOR]

This paper investigates the Poisson (co)homology of affine Poisson algebras. It is shown that there is a twisted Poincare duality between their Poisson homology and cohomology. The relation between the Poisson (co)homology of an affine Poisson algebra and the Hochschild (co)homology of its deformation quantization is also discussed, which is similar to Kassel's result (1988) for homology and is a special case of Kontsevich's theorem (2003) for cohomology. [ABSTRACT FROM AUTHOR]

The rational homology groups of the matching complexes are closely related to the syzygies of the Veronese embeddings. In this paper we will prove the vanishing of certain rational homology groups of matching complexes, thus proving that the third Veronese embeddings satisfy the property N6. This settles the Ottaviani-Paoletti conjecture for third Veronese embeddings. This result is optimal since V3 (Pn) does not satisfy the property N7 for n > 2 as shown by Ottaviani-Paoletti. [ABSTRACT FROM AUTHOR]

We show that the space RpRc(Wg2n) of metrics with positive Ricci curvature on the manifold Wg2n := ♯g (Sn × Sn) has nontrivial rational homology if n ≢ 3 (mod 4) and g are both sufficiently large. The same argument applies to RpRc(Wg2n ♯ N) provided that N is spin and Wg2n ♯ N admits a Ricci positive metric. [ABSTRACT FROM AUTHOR]

The well-rounded retract for SL_n(\mathbb {Z}) is defined as the set of flat tori of unit volume and dimension n whose systoles generate a finite-index subgroup in homology. This set forms an equivariant spine of minimal dimension for the space of flat tori. For both the Outer space X_n of metric graphs of rank n and the Teichmüller space \mathcal {T}_g of closed hyperbolic surfaces of genus g, we show that the literal analogue of the well-rounded retract does not contain an equivariant spine. We also prove that the sets of graphs whose systoles fill either topologically or geometrically (two analogues of a set proposed as a spine for \mathcal {T}_g by Thurston) are spines for X_n but that their dimension is larger than the virtual cohomological dimension of Out(F_n) in general. [ABSTRACT FROM AUTHOR]

We prove that the C2-equivariant Eilenberg-MacLane spectrum associated with the constant Mackey functor F2 is equivalent to a Thom spectrum over ΩρSρ+1. [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 prove that in the d-dimensional Linial–Meshulam stochastic process the (d - 1)st homology group with integer coefficients vanishes exactly when the final maximal (d - 1)-dimensional face is covered by a top-dimensional face. This generalizes the d = 2 case proved recently by Łuczak and Peled [Discrete Comput. Geom. 59 (2018), pp. 131–142] and establishes that p =d \log n/n is the sharp threshold for homology with integer coefficients to vanish in Y_d(n, p), answering a 2003 question of Linial and Meshulam [Combinatorica 26 (2006), pp. 475–487]. [ABSTRACT FROM AUTHOR]

Let k be a commutative Noetherian ring and let C be a locally finite k-linear category equipped with a self-embedding functor of degree 1. We show under a moderate condition that finitely generated torsion representations of C are super finitely presented (that is, they have projective resolutions, each term of which is finitely generated). In the situation that these self-embedding functors are genetic functors, we give upper bounds for homological degrees of finitely generated torsion modules. These results apply to quite a few categories recently appearing in representation stability theory. In particular, when k is a field of characteristic 0, using the result of Church and Ellenberg [arXiv:1506.01022], we obtain another upper bound for homological degrees of finitely generated FI-modules. [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]

The article focuses on the uniquness of centric linking systems for saturated fusion systems. It mentions that the open question in the theory of fusion systems with increasingly vibrant period of research. It offers information on the fusion system FS(G) of a finite group G with Sylow p-subgroup S.

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]

Racks and quandles are fundamental algebraic structures related to the topology of knots, braids, and the Yang-Baxter equation. We show that the cohomology groups usually associated with racks and quandles agree with the Quillen cohomology groups for the algebraic theories of racks and quandles, respectively. This makes available the entire range of tools that comes with a Quillen homology theory, such as long exact sequences (transitivity) and excision isomorphisms (flat base change). [ABSTRACT FROM AUTHOR]

The p-local homotopy types of gauge groups of principal G-bundles over S4 are classified when G is a compact connected exceptional Lie group without p-torsion in homology except for (G, p) = (E7, 5). [ABSTRACT FROM AUTHOR]

We introduce the notion of a resolution supported on a poset. When the poset is a CW-poset, i.e., the face poset of a regular CW-complex, we recover the notion of cellular resolution as introduced by Bayer and Sturmfels. Work of Reiner and Welker, and of Velasco, has shown that there are monomial ideals whose minimal free resolutions are not cellular, hence cannot be supported on any CW-poset. We show that for any monomial ideal there is a homology CW-poset that supports a minimal free resolution of the ideal. This allows one to extend to every minimal resolution, essentially verbatim, techniques initially developed to study cellular resolutions. As two demonstrations of this process, we show that minimal resolutions of toric rings are supported on what we call toric hcw-posets, and, generalizing results of Miller and Sturmfels, we prove a fundamental relationship between Artinianizations and Alexander duality for monomial ideals. [ABSTRACT FROM AUTHOR]

We show that the problem of recognizing that a knot diagram represents a specific torus knot, or any torus knot at all, is in the complexity class NP n co-NP, assuming the generalized Riemann hypothesis. We also show that satellite knot detection is in NP under the same assumption and that cabled knot detection and composite knot detection are unconditionally in NP. Our algorithms are based on recent work of Kuperberg and of Lackenby on detecting knottedness. [ABSTRACT FROM AUTHOR]

For any perfect fibration E → B, there is a “free loop transfer map” LB+ → LE+, defined using topological Hochschild homology. We prove that this transfer is compatible with the Becker-Gottlieb transfer, allowing us to extend a result of Dorabiała and Johnson on the transfer map in Waldhausen’s A-theory. In the case where E → B is a smooth fiber bundle, we also give a concrete geometric model for the free loop transfer in terms of Pontryagin-Thom collapse maps. We recover the previously known computations of the free loop transfer due to Schlichtkrull and make a few new computations as well. [ABSTRACT FROM AUTHOR]

We consider the topology and dynamics associated with a wide class of matchbox manifolds, including spaces of aperiodic tilings and suspensions of higher rank (potentially nonabelian) group actions on zero-dimensional spaces. For such a space we introduce a topological invariant, the homology core, built using an expansion of it as an inverse sequence of simplicial complexes. The invariant takes the form of a monoid equipped with a representation, which in many cases can be used to obtain a finer classification than is possible with the previously developed invariants. When the space is obtained by suspending a topologically transitive action of the fundamental group G of a closed orientable manifold on a zero-dimensional compact space Z, this invariant corresponds to the space of finite Borel measures on Z which are invariant under the action of G. This leads to connections between the rank of the core and the number of invariant, ergodic Borel probability measures for such actions. We illustrate with several examples how these invariants can be calculated and used for topological classification and how it leads to an understanding of the invariant measures. [ABSTRACT FROM AUTHOR]

In this note we show that a p-complete nilpotent space X has a p-complete suspension spectrum if and only if its homotopy groups p*X are bounded p-torsion. In contrast, if p*X is not all bounded p-torsion, we locate uncountable rational vector spaces in the integral homology and in the stable homotopy groups of X. To prove this, we establish a homological criterion for p-completeness of connective spectra. Moreover, we illustrate our results by studying the stable homotopy groups of K(Zp, n) via Goodwillie calculus. [ABSTRACT FROM AUTHOR]

For a Lie group G and a smooth manifold W, we study the difference between smooth actions of G on W and fiber bundles over the classifying space of G with fiber W and structure group Diff(W). In particular, we exhibit smooth manifold bundles over BSU(2) that are not induced by an action. The main tool for reaching this goal is a technical result that gives a constraint for the values of tautological classes of the fiber bundle associated to a group action. [ABSTRACT FROM AUTHOR]

Given a rational homology sphere which bounds rational homology balls, we investigate the complexity of these balls as measured by the number of 1-handles in a handle decomposition. We use Casson-Gordon invariants to obtain lower bounds which also lead to lower bounds on the fusion number of ribbon knots. We use Levine-Tristram signatures to compute these bounds and produce explicit examples. [ABSTRACT FROM AUTHOR]

In accordance with the Bing-Borsuk conjecture, we show that if X is an n-dimensional homogeneous metric ANR-compactum and x ∈ X, then there is a local basis at x consisting of connected open sets U such that the homological properties of U and bd U are similar to the properties of the closed ball Bn ⊂ Rn and its boundary Sn-1. We discuss also the following questions raised by Bing-Borsuk [Ann. of Math. (2) 81 (1965), 100–111], where X is a homogeneous ANR-compactum with dimX = n: • Is it true that X is cyclic in dimension n? • Is it true that no non-empty closed subset of X, acyclic in dimension n − 1, separates X? It is shown that both questions simultaneously have positive or negative answers, and a positive solution to each one of them implies a solution to another question of Bing-Borsuk (whether every finite-dimensional homogenous metric AR-compactum is a point). [ABSTRACT FROM AUTHOR]

We introduce an sln homology theory for knots and links in the thickened annulus. To do so, we first give a fresh perspective on sutured annular Khovanov homology, showing that its definition follows naturally from trace decategorifications of enhanced sl2 foams and categorified quantum glm, via classical skew Howe duality. This framework then extends to give our annular sln link homology theory, which we call sutured annular Khovanov-Rozansky homology. We show that the sln sutured annular Khovanov-Rozansky homology of an annular link carries an action of the Lie algebra sln, which in the n = 2 case recovers a result of Grigsby-Licata-Wehrli. [ABSTRACT FROM AUTHOR]

As an application of Behrens and Rezk’s spectral algebra model for unstable vn-periodic homotopy theory, we give explicit presentations for the completed E-homology of the Bousfield-Kuhn functor on odd-dimensional spheres at chromatic level 2, and compare them to the level 1 case. The latter reflects earlier work in the literature on K-theory localizations. [ABSTRACT FROM AUTHOR]

Knot contact homology is an invariant of knots derived from Legendrian contact homology which has numerous connections to the knot group. We use basic properties of knot groups to prove that knot contact homology detects every torus knot. Further, if the knot contact homology of a knot is isomorphic to that of a cable (respectively composite) knot, then the knot is a cable (respectively composite). [ABSTRACT FROM AUTHOR]

We analyze stable homology over associative rings and obtain results over Artin algebras and commutative noetherian rings. Our study develops similarly for these classes; for simplicity we only discuss the latter here. Stable homology is a broad generalization of Tate homology. Vanishing of stable homology detects classes of rings--among them Gorenstein rings, the original domain of Tate homology. Closely related to Gorensteinness of rings is Auslander's G-dimension for modules. We show that vanishing of stable homology detects modules of finite G-dimension. This is the first characterization of such modules in terms of vanishing of (co)homology alone. Stable homology, like absolute homology, Tor, is a theory in two variables. It can be computed from a flat resolution of one module together with an injective resolution of the other. This betrays that stable homology is not balanced in the way Tor is balanced. In fact, we prove that a ring is Gorenstein if and only if stable homology is balanced. [ABSTRACT FROM AUTHOR]

The relationships between the homological properties and the invariants of I, Gin(I) and Ilex have been studied extensively over the past decades. A result of A. Conca, J. Herzog and T. Hibi points out some rigid behaviours of their Betti numbers. In this work we establish a local cohomology counterpart of their theorem. To this end, we make use of properties of sequentially Cohen-Macaulay modules and we study a generalization of such concept by introducing what we call partially sequentially Cohen-Macaulay modules, which might be of interest by themselves. [ABSTRACT FROM AUTHOR]

In this note, we show that if a pseudo-Anosov map φ : S → S admits a finite cover whose action on the first homology has spectral radius greater than 1, then the monodromy of any fibered structure of any finite cover of the mapping torus Mφ has the same property. [ABSTRACT FROM AUTHOR]

Using a combination of Atiyah-Segal ideas on one side and of Connes and Baum-Connes ideas on the other, we prove that the twisted geometric K-homology groups of a Lie groupoid have an external multiplicative structure extending hence the external product structures for proper cases considered by Adem-Ruan in 2003 or by Tu, Xu and Laurent-Gengoux in 2004. These twisted geometric K-homology groups are the left-hand sides of the twisted geometric Baum-Connes assembly maps recently constructed by Carrillo Rouse and Wang (2016), and hence one can transfer the multiplicative structure via the Baum-Connes map to the twisted K-theory groups whenever these assembly maps are isomorphisms. [ABSTRACT FROM AUTHOR]

For a Cohen-Macaulay ring R, we exhibit the equivalence of the bounded derived categories of certain resolving subcategories, which, amongst other results, yields an equivalence of the bounded derived category of finite length and finite projective dimension modules with the bounded derived category of projective modules with finite length homologies. This yields isomorphisms of K-theory and Witt groups (amongst other invariants) and improves on terms of associated spectral sequences and Gersten complexes. [ABSTRACT FROM AUTHOR]

We prove that if an algebra is either selfinjective, local or graded, then the Hochschild homology dimension of its trivial extension is infinite. [ABSTRACT FROM AUTHOR]

We calculate the intersection ring of 3-dimensional graph manifolds with rational coefficients and give an algebraic characterization of these rings when the manifold's underlying graph is a tree. We are able to use this characterization to show that the intersection ring obstructs arbitrary 3-manifolds from being homology cobordant to certain graph manifolds. [ABSTRACT FROM AUTHOR]

The splitting number of a link is the minimal number of crossing changes between different components required to convert it into a split link. We obtain a lower bound on the splitting number in terms of the (multivariable) signature and nullity. Although very elementary and easy to compute, this bound turns out to be suprisingly efficient. In particular, it makes it a routine check to recover the splitting number of 129 out of the 130 prime links with at most 9 crossings. Also, we easily determine 16 of the 17 splitting numbers that were studied by Batson and Seed using Khovanov homology, and later computed by Cha, Friedl and Powell using a variety of techniques. Finally, we determine the splitting number of a large class of 2-bridge links which includes examples recently computed by Borodzik and Gorsky using a Heegaard Floer theoretical criterion. [ABSTRACT FROM AUTHOR]

In this note we study the homology of configuration spaces of closed manifolds. We sharpen the eventual periodicity results of Nagpal and Cantero-Palmer, prove integral homological stability for configuration spaces of odd-dimensional manifolds and introduce a stabilization map on the homology with Z[1/2]-coefficients of configuration spaces of odd-dimensional manifolds. [ABSTRACT FROM AUTHOR]