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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]

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]

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]

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]

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]

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 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]

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]

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 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]

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]

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]

In this paper we prove (integral) homological stability for the sequences of spaces C+n (M,X). These are the spaces of configurations of n points in a connected manifold of dimension at least 2 which 'admits a boundary', with labels in a path-connected space X, and with an orientation -- an ordering of the points up to even permutations. They are double covers of the unordered configuration spaces Cn(M,X), and indeed to prove our result we adapt methods from a paper of Randal- Williams, which proves homological stability in the unordered case. Interestingly the oriented configuration spaces stabilise more slowly than the unordered ones: the stability slope we obtain is ⅓, compared to ½ in the unordered case (and these are the best possible slopes in their respective cases). This result can also be interpreted as homological stability for the unordered configuration spaces with certain twisted Z ⊕ Z-coefficients. [ABSTRACT FROM AUTHOR]

In this paper we determine the integral quandle homology groups of Alexander quandles of prime order. As a special case, this settles the delayed Fibonacci conjecture by M. Niebrzydowski and J. H. Przytycki from their 2009 paper. Further, we determine the cohomology group of the Alexander quandle and obtain relatively simple presentations of all higher degree cocycles which generate the cohomology group. Finally, we prove that the integral quandle homology of a finite connected Alexander quandle is annihilated by the order of the quandle. [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]

In recent work, Beukers characterized A-hypergeometric systems having a full set of algebraic solutions. He accomplished this by (1) determining which A-hypergeometric systems have a full set of polynomial solutions modulo p for almost all primes p and (2) showing that these systems come from geometry. He then applied a fundamental theorem of N. Katz, which says that such systems have a full set of algebraic solutions. In this paper we establish some connections between nonresonant A-hypergeometric systems and de Rham-type complexes, which leads to a determination of which Ahypergeometric systems come from geometry. We do not use the fact that the system is irreducible or find integral formulas for its solutions. [ABSTRACT FROM AUTHOR]

R. Sazeedeh showed that top local cohomology modules are Gorenstein injective in a Gorenstein local ring with at most two dimensions. In this paper, it is proved that the condition of dimension in his result cannot be relaxed and the conclusion in his result holds for complete local hypersurface rings with any dimension. [ABSTRACT FROM AUTHOR]

Let B be the ring of bounded operators in a complex, separable Hilbert space. For p > 0 consider the Schatten ideal LP consisting of those operators whose sequence of singular values is p-summable; put S =UpLp Let G be a group and Vcyc the family of virtually cyclic subgroups. Guoliang Yu proved that the K-theory assembly mapH*G(E(G, Vcyc),K(S)) → K*(S[G]) is rationally injective. His proof involves the construction of a certain Chern character tailored to work with coefficients S and the use of some results about algebraic K-theory of operator ideals and about controlled topology and coarse geometry. In this paper we give a different proof of Yu's result. Our proof uses the usual Chern character to cyclic homology. Like Yu's, it relies on results on algebraic K-theory of operator ideals, but no controlled topology or coarse geometry techniques are used. We formulate the result in terms of homotopy K-theory. We prove that the rational assembly map HG*(E(G,Fin),KH(Lp)) ⊗ Q... → KH*(Lp[G]) ⊗ Q... is injective. We show that the latter map is equivalent to the assembly map considered by Yu, and thus obtain his result as a corollary. [ABSTRACT FROM AUTHOR]

Let S be a compact oriented surface of genus g with one boundary component. Homology cylinders over S form a monoid IC into which the Torelli group I of S embeds by the mapping cylinder construction. Two homology cylinders M and M' are said to be Yk-equivalent if M' is obtained from M by "twisting" an arbitrary surface S ⊂ M with a homeomorphism belonging to the k-th term of the lower central series of the Torelli group of S. The Jk-equivalence relation on IC is defined in a similar way using the k-th term of the Johnson filtration. In this paper, we characterize the Y3- equivalence with three classical invariants: (1) the action on the third nilpotent quotient of the fundamental group of S, (2) the quadratic part of the relative Alexander polynomial, and (3) a by-product of the Casson invariant. Similarly, we show that the J3-equivalence is classified by (1) and (2). We also prove that the core of the Casson invariant (originally defined by Morita on the second term of the Johnson filtration of I) has a unique extension (to the corresponding submonoid of IC) that is preserved by Y3-equivalence and the mapping class group action. [ABSTRACT FROM AUTHOR]

In this expository paper we introduce extended topological quantum field theories and the cobordism hypothesis. [ABSTRACT FROM AUTHOR]

A projective scheme X is called 'quadratic' if X is scheme-theoretically cut out by homogeneous equations of degree 2. Furthermore, we say that X satisfies 'property N2,p' if it is quadratic and the quadratic ideal has only linear syzygies up to the first p-th steps. In the present paper, we compare the linear syzygies of the inner projections with those of X and obtain a theorem on 'embedded linear syzygies' as one of our main results. This is the natural projection-analogue of 'restricting linear syzygies' in the linear section case. As an immediate corollary, we show that the inner projections of X satisfy property N2,p-1 for any reduced scheme X with property N2,p. Moreover, we also obtain the neccessary lower bound (codim X)·p - p(p-1)― 2, which is sharp, on the number of quadrics vanishing on X in order to satisfy N2,p and show that the arithmetic depths of inner projections are equal to that of the quadratic scheme X. These results admit an interesting 'syzygetic' rigidity theorem on property N2,p which leads the classifications of extremal and next-to-extremal cases. For these results we develop the elimination mapping cone theorem for infinitely generated graded modules and improve the partial elimination ideal theory initiated by M. Green. This new method allows us to treat a wider class of projective schemes which cannot be covered by the Koszul cohomology techniques because these are not projectively normal in general. [ABSTRACT FROM AUTHOR]

In this paper, we show that rational homology 3-spheres are ubiquitous from the viewpoint of Heegaard splitting. Let M = H+ UF H_ be a genus g Heegaa;rd splitting of a closed 3-manifold and c be a simple closed curve in F. Then there is a 3-manifold Mc which is obtained from M by horizontal Dehn surgery along c. We show that for c such that the homology class [c] is generic in the set of curve-represented homology classes Hs ⊂ H1(F), rank(H1(Mc,Q)) < max{ 1, rank(H1(M,Q)}. As a corollary, for a set of cusub>, c2, hellip;, cm}, m ≥ g, such that each [ci] is generic in Hs ⊂ H1(F), M(c1,c2,…,cm) is a rational homology 3-sphere. [ABSTRACT FROM AUTHOR]

For a given endomorphism on a unitary k-algebra, A, with k in the center of A, there are definitions of twisted cyclic and Hochschild homology. This paper will show that the method used to define them can be used to define twisted de Rham homology. The main result is that twisted de Rham homology can be thought of as the kernel of the Connes map from twisted cyclic homology to twisted Hochschild homology. [ABSTRACT FROM AUTHOR]

Transfer maps and projection formulas are undoubtedly one of the key tools in the development and computation of (co)homology theories. In this paper we develop a unified treatment of transfer maps and projection formulas in the non-commutative setting of dg categories. As an application, we obtain transfer maps and projection formulas in algebraic K-theory, cyclic homology, topological cyclic homology, and other scheme invariants. [ABSTRACT FROM AUTHOR]

Cohomological induction gives an algebraic method for constructing representations of a real reductive Lie group G from irreducible representations of reductive subgroups. Beilinson-Bernstein localization alternatively gives a geometric method for constructing Harish-Chandra modules for G from certain representations of a Cartan subgroup. The duality theorem of Hecht, Mil&icaron;cíc, Schmid and Wolf establishes a relationship between modules cohomologically induced from minimal parabolics and the cohomology of the D-modules on the complex flag variety for G determined by the Beilinson- Bernstein construction. The main results of this paper give a generalization of the duality theorem to partial flag varieties, which recovers cohomologically induced modules arising from nonminimal parabolics. [ABSTRACT FROM AUTHOR]

The decorated hypercube found in the construction of Khovanov homology for links is an example of a Boolean lattice equipped with a presheaf of modules. One can place this in a wider setting as an example of a coloured poset, that is to say, a poset with a unique maximal element equipped with a presheaf of modules. In this paper we initiate the study of a bundle theory for coloured posets, producing for a certain class of base posets a Leray-Serre type spectral sequence. We then show how this theory finds an application in Khovanov homology by producing a new spectral sequence converging to the Khovanov homology of a given link. [ABSTRACT FROM AUTHOR]

A degeneration of compact Kähler manifolds gives rise to a monodromy action on the Betti moduli space H¹(X, G) = Hom(π1(x), G)/G over smooth fibres with a complex algebraic structure group G that is either abelian or reductive. Assume that the singularities of the central fibre are of normal crossing. When G = C, the invariant cohomology classes arise from the global classes. This is no longer true in general. In this paper, we produce large families of locally invariant classes that do not arise from global ones for reductive G. These examples exist even when G is abelian as long as G contains multiple torsion points. Finally, for general G, we make a new conjecture on local invariant classes and produce some suggestive examples. [ABSTRACT FROM AUTHOR]

We define degeneracy loci for vector bundles with structure group G2 and give formulas for their cohomology (or Chow) classes in terms of the Chern classes of the bundles involved. When the base is a point, such formulas are part of the theory for rational homogeneous spaces developed by Bernstein-Gelfand-Gelfand and Demazure. This has been extended to the setting of general algebraic geometry by Giambelli-Thom-Porteous, Kempf-Laksov, and Fulton in classical types; the present work carries out the analogous program in type G2. We include explicit descriptions of the G2 flag variety and its Schubert varieties, and several computations, including one that answers a question of W. Graham. In appendices, we collect some facts from representation theory and compute the Chow rings of quadric bundles, correcting an error in a paper by Edidin and Graham. [ABSTRACT FROM AUTHOR]

First cohomology groups of finite groups with nontrivial irreducible coefficients have been useful in several geometric and arithmetic contexts, including Wiles''s famous paper (1995). Internal to group theory, $1$-cohomology plays a role in the general theory of maximal subgroups of finite groups, as developed by Aschbacher and Scott (1985).One can pass to the case where the group acts faithfully and the underlying module is absolutely irreducible. In this case, R. Guralnick (1986) conjectured that there is a universal constant bounding all of the dimensions of these cohomology groups. This paper provides the first general positive results on this conjecture, proving that the generic 1-cohomology $H^1_{operatorname {gen}}(G,L) :=underset {qto infty }{lim } H^1(G(q),L)$ (see Cline, Parshall, Scott, and van der Kallen) (1977) of a finite group $G(q)$ of Lie type, with absolutely irreducible coefficients $L$ (in the defining characteristic of $G$), is bounded by a constant depending only on the root system. In all cases, we are able to improve this result to a bound on $H^1(G(q),L)$ itself, still depending only on the root system. The generic $H^1$ result, and related results for $operatorname {Ext}^1$, emerge here as a consequence of a general study, of interest in its own right, of the homological properties of certain rational modules $Delta ^{text {rm red}}(lambda ), nabla _{text {rm red}}(lambda )$, indexed by dominant weights $lambda $, for a reductive group $G$. The modules $Delta ^{text {rm red}}(lambda )$ and $nabla _{text {rm red}}(lambda )$ arise naturally from irreducible representations of the quantum enveloping algebra $U_zeta $ (of the same type as $G$) at a $p$th root of unity, where $p>0$ is the characteristic of the defining field for $G$. Finally, we apply our $operatorname {Ext}^1$-bounds, results of Bendel, Nakano, and Pillen (2006), as well as results of Sin (1993), (1992), (1994) on the Ree and Suzuki groups to obtain the (non-generic) bounds on $H^1(G(q),L)$. [ABSTRACT FROM AUTHOR]

This paper records two results which were inexplicably omitted from the paper on Pin structures on low dimensional manifolds in the LMS Lecture Note Series, volume 151, by Kirby and this author. Kirby declined to be listed as a coauthor of this paper. par A $Pin^{-}$-structure on a surface $X$ induces a quadratic enhancement of the mod $2$ intersection form, $qcolon H_1(X;mathbb {Z}/2mathbb {Z})to mathbb {Z}/4mathbb {Z}$. par Theorem 1.1 says that $q$ vanishes on the kernel of the map in homology to a bounding $3$-manifold. This is used by Kreck and Puppe in their paper in Homology, Homotopy and Applications, volume 10. The arXiv version, arXiv:0707.1599 [math.AT], referred to an email from the author to Kreck for the proof. A more polished and public proof seems desirable. par In Section 6 of the paper with Kirby, a $Pin^{-}$-structure is constructed on a surface $X$ dual to $w_2$ in an oriented 4-manifold, $M^4$. Theorem 2.1 says that $q$ vanishes on the Poincaré dual to the image of $H^1(M;mathbb {Z}/2mathbb {Z})$ in $H^1(X;mathbb {Z}/2mathbb {Z})$. [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]