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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]

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]

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]

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]

The $i$-th local cohomology module of a finitely generated graded module $M$ over a standard positively graded commutative Noetherian ring $R$, with respect to the irrelevant ideal $R_+$, is itself graded; all its graded components are finitely generated modules over $R_0$, the component of $R$ of degree $0$. It is known that the $n$-th component $H^i_{R_+}(M)_n$ of this local cohomology module $H^i_{R_+}(M)$ is zero for all $n>> 0$. This paper is concerned with the asymptotic behaviour of $\operatorname{Ass}_{R_0}(H^i_{R_+}(M)_n)$ as $n \rightarrow -\infty$. The smallest $i$ for which such study is interesting is the finiteness dimension $f$ of $M$ relative to $R_+$, defined as the least integer $j$ for which $H^j_{R_+}(M)$ is not finitely generated. Brodmann and Hellus have shown that $\operatorname{Ass}_{R_0}(H^f_{R_+}(M)_n)$ is constant for all $n < < 0$ (that is, in their terminology, $\operatorname{Ass}_{R_0}(H^f_{R_+}(M)_n)$ is asymptotically stable for $n \rightarrow -\infty$). The first main aim of this paper is to identify the ultimate constant value (under the mild assumption that $R$ is a homomorphic image of a regular ring): our answer is precisely the set of contractions to $R_0$ of certain relevant primes of $R$ whose existence is confirmed by Grothendieck's Finiteness Theorem for local cohomology. Brodmann and Hellus raised various questions about such asymptotic behaviour when $i > f$. They noted that Singh's study of a particular example (in which $f = 2$) shows that $\operatorname{Ass}_{R_0}(H^3_{R_+}(R)_n)$ need not be asymptotically stable for $n \rightarrow -\infty$. The second main aim of this paper is to determine, for Singh's example, $\operatorname{Ass}_{R_0}(H^3_{R_+}(R)_n)$ quite precisely for every integer $n$, and, thereby, answer one of the questions raised by Brodmann and... [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]

In this paper we introduce a new elliptic complex on an odd-dimensional manifold with a self-dual line field. The notion of a self-dual line field is a generalization of the notion of a conformal line field. Ellipticity, Fredholm properties and Hodge decompositions of these new complexes are proved both in the case of a closed manifold and in the case of a manifold with boundary. The cohomology groups of these elliptic complexes are computed in some cases. In addition, in this paper, we generalize the notion of an anti-self-dual connection on a smooth 4-manifold to a 3-manifold with a line field and a smooth 5-manifold with a line field. The above new elliptic complexes can be twisted by anti-self-dual connections in dimensions 3 and 5, but only by flat connections in dimensions above 5. This reveals a special feature of dimensions 3 and 5. [ABSTRACT FROM AUTHOR]

The main result of the first part of the paper is a generalization of the classical result of Menger-Urysohn : $\dim (A\cup B)\le \dim A+\dim B+1$. \begin{theorem1} Suppose $A,B$ are subsets of a metrizable space and $K$ and $L$ are CW complexes. If $K$ is an absolute extensor for $A$ and $L$ is an absolute extensor for $B$, then the join $K*L$ is an absolute extensor for $A\cup B$. \end{theorem1} As an application we prove the following analogue of the Menger-Urysohn Theorem for cohomological dimension: \begin{theorem2} Suppose $A,B$ are subsets of a metrizable space. Then \begin{equation*}\dim _{R }(A\cup B)\le \dim _{R }A+\dim _{R }B+1 \end{equation*} for any ring $R $ with unity and \begin{equation*}\dim _{G}(A\cup B)\le \dim _{G}A+\dim _{G}B+2\end{equation*} for any abelian group $G$. \end{theorem2} The second part of the paper is devoted to the question of existence of universal spaces: \begin{theorem3} Suppose $\{K_{i}\}_{i\ge 1}$ is a sequence of CW complexes homotopy dominated by finite CW complexes. Then \begin{itemize} \item[a.] Given a separable, metrizable space $Y$ such that $K_{i}\in AE(Y)$, $i\ge 1$, there exists a metrizable compactification $c(Y)$ of $Y$ such that $K_{i}\in AE(c(Y))$, $i\ge 1$. \item[b.] There is a universal space of the class of all compact metrizable spaces $Y$ such that $K_{i}\in AE(Y)$ for all $i\ge 1$. \item[c.] There is a completely metrizable and separable space $Z$ such that $ K_{i}\in AE(Z)$ for all $i\ge 1$ with the property that any completely metrizable and separable space $Z'$ with $ K_{i}\in AE(Z')$ for all $i\ge 1$ embeds in $Z$ as a closed subset. \end{itemize}\end{theorem3} [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]

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]

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]

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]

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]

The purpose of this paper is twofold: first, we extend Saito's filtration on Chow groups, which is a candidate for the conjectural Bloch Beilinson filtration on the Chow groups of a smooth projective variety, from Chow groups to the bivariant Chow groups. In order to do this, we construct cycle class maps from the bivariant Chow groups to bivariant cohomology groups. Secondly, we use our methods to define a bivariant version of Bloch's higher Chow groups. [ABSTRACT FROM AUTHOR]

In this paper we define the 1,2-coloured HOMFLY-PT triply graded link homology and prove that it is a link invariant. We also conjecture on how to generalize our construction for arbitrary colours. [ABSTRACT FROM AUTHOR]

In this paper, we compute the Gerstenhaber bracket on the Hoch-schild cohomology of $C^infty (M)rtimes G$ for a finite group $G$ acting on a compact manifold $M$. Using this computation, we obtain geometric descriptions for all noncommutative Poisson structures on $C^infty (M)rtimes G$ when $M$ is a symplectic manifold. We also discuss examples of deformation quantizations of these noncommutative Poisson structures. [ABSTRACT FROM AUTHOR]

In this paper, we determine the cokernel of the $k$-th Johnson homomorphisms of the automorphism group of a free metabelian group for $k geq 2$ and $n geq 4$. As a corollary, we obtain a lower bound on the rank of the graded quotient of the Johnson filtration of the automorphism group of a free group. Furthermore, by using the second Johnson homomorphism, we determine the image of the cup product map in the rational second cohomology group of the IA-automorphism group of a free metabelian group, and show that it is isomorphic to that of the IA-automorphism group of a free group which is already determined by Pettet. Finally, by considering the kernel of the Magnus representations of the automorphism group of a free group and a free metabelian group, we show that there are non-trivial rational second cohomology classes of the IA-automorphism group of a free metabelian group which are not in the image of the cup product map. [ABSTRACT FROM AUTHOR]

In this paper, we give a combinatorial definition of a higher homotopy commutativity of the multiplication for an $A_n$-space. To give the definition, we use polyhedra called the permuto-associahedra which are constructed by Kapranov. We also show that if a connected $A_p$-space has the finitely generated mod $p$ cohomology for a prime $p$ and the multiplication of it is homotopy commutative of the $p$-th order, then it has the mod $p$ homotopy type of a finite product of Eilenberg-Mac Lane spaces $K(\mathbb{Z},1)$s, $K(\mathbb{Z},2)$s and $K(\mathbb{Z}/p^i,1)$s for $i\ge 1$. [ABSTRACT FROM AUTHOR]

It is well known that the Hochschild cohomology $H^*(A,A)$ of an associative algebra $A$ admits a G-algebra structure. In this paper we show that the dialgebra cohomology $HY^*(D,D)$ of an associative dialgebra $D$ has a similar structure, which is induced from a homotopy G-algebra structure on the dialgebra cochain complex $CY^*(D,D)$. [ABSTRACT FROM AUTHOR]

The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. A proof is given in this paper that this sphere is distinct from the same sphere with its orientation reversed. Our proof is based on a state-sum invariant for knotted surfaces developed via a cohomology theory of racks and quandles (also known as distributive groupoids). \par A quandle is a set with a binary operation --- the axioms of which model the Reidemeister moves in classical knot theory. Colorings of diagrams of knotted curves and surfaces by quandle elements, together with cocycles of quandles, are used to define state-sum invariants for knotted circles in $3$-space and knotted surfaces in $4$-space. \par Cohomology groups of various quandles are computed herein and applied to the study of the state-sum invariants. Non-triviality of the invariants is proved for a variety of knots and links, and conversely, knot invariants are used to prove non-triviality of cohomology for a variety of quandles. [ABSTRACT FROM AUTHOR]

We investigate algebraic $\Gamma $-monomials of Thakur's positive characteristic $\Gamma $-function, by using Anderson and Das' double complex method of computing the sign cohomology of the universal ordinary distribution. We prove that the $\Gamma $-monomial associated to an element of the second sign cohomology of the universal ordinary distribution of $\mathbb{F}_{q}(T)$ generates a Kummer extension of some Carlitz cyclotomic function field, which is also a Galois extension of the base field $\mathbb{F}_{q}(T)$. These results are characteristic-$p$ analogues of those of Deligne on classical $\Gamma $-monomials, proofs of which were given by Das using the double complex method. In this paper, we also obtain some results on $e$-monomials of Carlitz's exponential function. [ABSTRACT FROM AUTHOR]

In [5], we presented a discrete Morse Theory that can be applied to general cell complexes. In particular, we defined the notion of a discrete Morse function, along with its associated set of critical cells. We also constructed a discrete Morse cocomplex, built from the critical cells and the gradient paths between them, which has the same cohomology as the underlying cell complex. In this paper we show how various cohomological operations are induced by maps between Morse cocomplexes. For example, given three discrete Morse functions, we construct a map from the tensor product of the first two Morse cocomplexes to the third Morse cocomplex which induces the cup product on cohomology. All maps are constructed by counting certain configurations of gradient paths. This work is closely related to the corresponding formulas in the smooth category as presented by Betz and Cohen [2] and Fukaya [11], [12]. [ABSTRACT FROM AUTHOR]

This paper concerns the rational cohomology ring of the complement $M$ of a complex subspace arrangement. We start with the De Concini-Procesi differential graded algebra that is a rational model for $M$. Inside it we find a much smaller subalgebra $D$ quasi-isomorphic to the whole algebra. $D$ is described by defining a natural multiplication on a chain complex whose homology is the local homology of the intersection lattice $L$ whence connecting the De Concini-Procesi model with the Goresky-MacPherson formula for the additive structure of $H^*(M)$. The algebra $D$ has a natural integral version that is a good candidate for an integral model of $M$. If the rational local homology of $L$ can be computed explicitly we obtain an explicit presentation of the ring $H^*(M,{\mathbf Q})$. For example, this is done for the cases where $L$ is a geometric lattice and where $M$ is a $k$-equal manifold. [ABSTRACT FROM AUTHOR]

Let $p$ be an odd prime number and let $G$ be an extraspecial $p$-group. The purpose of the paper is to show that $G$ has no non-zero essential mod-$p$ cohomology (and in fact that $H^{*}(G,\mathbb{F}_{p})$ is Cohen-Macaulay) if and only if $|G|=27$ and $exp(G)=3$. [ABSTRACT FROM AUTHOR]

Part I of this paper describes power operations in elliptic cohomology in terms of isogenies of the underlying elliptic curve. Part II discusses a relationship between equivariant elliptic cohomology and representations of loop groups. Part III investigates the representation of theoretic considerations which give rise to the power operations discussed in Part I. [ABSTRACT FROM AUTHOR]

In this paper we study the set of periods of holomorphic maps on compact manifolds, using the periodic Lefschetz numbers introduced by Dold and Llibre, which can be computed from the homology class of the map. We show that these numbers contain information about the existence of periodic points of a given period; and, if we assume the map to be transversal, then they give us the exact number of such periodic orbits. We apply this result to the complex projective space of dimension $n$ and to some special type of Hopf surfaces, partially characterizing their set of periods. In the first case we also show that any holomorphic map of ${\mathbb CP}(n)$ of degree greater than one has infinitely many distinct periodic orbits, hence generalizing a theorem of Fornaess and Sibony. We then characterize the set of periods of a holomorphic map on the Riemann sphere, hence giving an alternative proof of Baker's theorem. [ABSTRACT FROM AUTHOR]

In this paper we show that, under a suitable condition, every nonsingular geometric flow on a manifold which is modeled on the Furstenberg boundary of $X$, where $X$ is a symmetric space of non-compact type, induces a torus action, and, in particular, if the manifold is a rational homology sphere, then the flow has a closed orbit. [ABSTRACT FROM AUTHOR]

Let $p$ be a prime number and $G$ be a compact Lie group. A homology decomposition for the classifying space $BG$ is a way of building $BG$ up to mod $p$ homology as a homotopy colimit of classifying spaces of subgroups of $G$. In this paper we develop techniques for constructing such homology decompositions. Jackowski, McClure and Oliver (\textit{Homotopy classification of self-maps of BG via $G$-actions}, Ann. of Math. \textbf{135} (1992), 183--270) construct a homology decomposition of $BG$ by classifying spaces of $p$-stubborn subgroups of $G$. Their decomposition is based on the existence of a finite-dimensional mod $p$ acyclic $G$-$CW$-complex with restricted set of orbit types. We apply our techniques to give a parallel proof of the $p$-stubborn decomposition of $BG$ which does not use this geometric construction. [ABSTRACT FROM AUTHOR]

In this paper we study finite group extensions represented by special cohomology classes. As an application, we obtain some restrictions on finite groups which can act freely on a product of spheres or on a product of real projective spaces. In particular, we prove that if $(Z/p)^r$ acts freely on $(S^1)^k$, then $r \leq k$. [ABSTRACT FROM AUTHOR]

In this paper we will study the cohomology of a family of $p$-groups associated to $\mathbb{F}_p$-Lie algebras. More precisely, we study a category $\mathbf{BGrp}$ of $p$-groups which will be equivalent to the category of $\mathbb{F}_p$-bracket algebras (Lie algebras minus the Jacobi identity). We then show that for a group $G$ in this category, its $\mathbb{F}_p$-cohomology is that of an elementary abelian $p$-group if and only if it is associated to a Lie algebra. We then proceed to study the exponent of $H^*(G ;\mathbb{Z})$ in the case that $G$ is associated to a Lie algebra $\mathfrak{L}$. To do this, we use the Bockstein spectral sequence and derive a formula that gives $B_2^*$ in terms of the Lie algebra cohomologies of $\mathfrak{L}$. We then expand some of these results to a wider category of $p$-groups. In particular, we calculate the cohomology of the $p$-groups $\Gamma_{n,k}$ which are defined to be the kernel of the mod $p$ reduction $ GL_n(\mathbb{Z}/p^{k+1}\mathbb{Z}) \overset{mod}{\longrightarrow} GL_n(\mathbb{F}_p). $ [ABSTRACT FROM AUTHOR]

This paper studies the homology of finite modules over the exterior algebra $E$ of a vector space $V$. To such a module $M$ we associate an algebraic set $V_E(M)\subseteq V$, consisting of those $v\in V$ that have a non-minimal annihilator in $M$. A cohomological description of its defining ideal leads, among other things, to complementary expressions for its dimension, linked by a `depth formula'. Explicit results are obtained for $M=E/J$, when $J$ is generated by products of elements of a basis $e_1,\dots,e_n$ of $V$. A (infinite) minimal free resolution of $E/J$ is constructed from a (finite) minimal resolution of $S/I$, where $I$ is the squarefree monomial ideal generated by `the same' products of the variables in the polynomial ring $S=K[x_1,\dots,x_n]$. It is proved that $V_E(E/J)$ is the union of the coordinate subspaces of $V$, spanned by subsets of $\{e_1,\dots,e_n\}$ determined by the Betti numbers of $S/I$ over $S$. [ABSTRACT FROM AUTHOR]

The problem 10 posed by Tudor Ganea is known as {\it the Ganea conjecture } on {\it a co-H-space}, a space with co-H-structure. Many efforts are devoted to show the Ganea conjecture under additional assumptions on the given co-H-structure. In this paper, we show a homological property of co-H-spaces in a slightly general situation. As a corollary, we get the Ganea conjecture for spaces up to dimension 3. [ABSTRACT FROM AUTHOR]

The purpose of the paper is to give an alternative construction and the proof of the main properties of symplectic invariants developed by Viterbo. Our approach is based on Morse homology theory. This is a step towards relating the ``finite dimensional'' symplectic invariants constructed via generating functions to the ``infinite dimensional'' ones constructed via Floer theory in Y.-G. Oh, \textit{Symplectic topology as the geometry of action functional}. I, J. Diff. Geom. \textbf{46} (1997), 499--577. [ABSTRACT FROM AUTHOR]

In this paper we construct an infinite dimensional (extraordinary) cohomology theory and a Morse theory corresponding to it. These theories have some special properties which make them useful in the study of critical points of strongly indefinite functionals (by strongly indefinite we mean a functional unbounded from below and from above on any subspace of finite codimension). Several applications are given to Hamiltonian systems, the one-dimensional wave equation (of vibrating string type) and systems of elliptic partial differential equations. [ABSTRACT FROM AUTHOR]

Given an associative algebra $A$ and the category $ \mathcal C$ of its finite dimensional modules, additional structures on the algebra $A$ induce corresponding ones on the category $\mathcal C$. Thus, the structure of a rigid quasi-tensor (braided monoidal) category on $Rep_{A}$ is induced by an algebra homomorphism $A\to A\otimes A$ (comultiplication), coassociative up to conjugation by $\Phi \in A^{\otimes 3}$ (associativity constraint) and cocommutative up to conjugation by $\mathcal R \in A^{\otimes 2}$ (commutativity constraint), together with an antiautomorphism (antipode) $S$ of $A$ satisfying the compatibility conditions. A morphism of quasi-tensor structures is given by an element $F\in A^{\otimes 2}$ with suitable induced actions on $\Phi $, $\mathcal R$ and $S$. Drinfeld defined such a structure on $A=U(\mathcal G )[[h]]$ for any semisimple Lie algebra $\mathcal{G} $ with the usual comultiplication and antipode but nontrivial $\mathcal R $ and $\Phi $, and proved that the corresponding quasi-tensor category is isomomorphic to the category of representations of the Drinfeld-Jimbo (DJ) quantum universal enveloping algebra (QUE), $U_{h}(\mathcal G )$. In the paper we give a direct cohomological construction of the $F$ which reduces $\Phi $ to the trivial associativity constraint, without any assumption on the prior existence of a strictly coassociative QUE. Thus we get a new approach to the DJ quantization. We prove that $F$ can be chosen to satisfy some additional invariance conditions under (anti)automorphisms of $U(\mathcal G )[[h]]$, in particular, $F$ gives an isomorphism of rigid quasi-tensor categories. Moreover, we prove that for pure imaginary values of the deformation parameter, the elements $F$, $R$ and $\Phi $ can be chosen to be formal unitary operators on the second and third tensor powers of the regular representation of the Lie group associated to $\mathcal G $ with $\Phi $ depending only on even powers of the deformation parameter. In addition, we consider some extra properties of these elements and give their interpretation in terms of additional structures on the relevant categories. [ABSTRACT FROM AUTHOR]

The paper is devoted to computation of the cohomology of the complex of logarithmic differential forms with coefficients in rational functions whose poles are located on the union of several hyperplanes of a linear space over a field of characteristic zero. The main result asserts that for a vast class of hyperplane arrangements, including all free and generic arrangements, the cohomology algebra coincides with the Orlik-Solomon algebra. Over the field of complex numbers, this means that the cohomologies coincide with the cohomologies of the complement of the union of the hyperplanes. We also prove that the cohomologies do not change if poles of arbitrary multiplicity are allowed on some of the hyperplanes. In particular, this gives an analogue of the algebraic de Rham theorem for an arbitrary arrangement over an arbitrary field of zero characteristic. [ABSTRACT FROM AUTHOR]

This paper deals with the local solvability of systems of first order linear partial differential equations defined by a germ $\omega $ at $0\in \mathbb{R}^{n+1}$ of a $\mathbb{C}$-valued, formally integrable ($\omega \wedge d\omega =0$), 1-form with nondegenerate Levi form. More precisely, the size of the obstruction to the solvability, for $(q-1)$-forms $u$, of the equation \begin{equation*}du\wedge \omega =\eta \wedge \omega ,\end{equation*} where $\eta $ is a given $q$-form satisfying $d\eta \wedge \omega =0$ is estimated in terms of the De Rham cohomology relative to $\omega $ [ABSTRACT FROM AUTHOR]

The filtrations on the James construction on spheres, $ J_{k}\left (S^{2n}\right )$, have played a major role in the study of the double suspension $S^{2n-1}\to \Omega ^2 S^{2n+1}$ and have been used to get information about the homotopy groups of spheres and Moore spaces and to construct product decompositions of related spaces. In this paper we calculate $ H_*\left ( \Omega J_{k}\left (S^{2n}\right ); {\mathbb{Z}}/p{\mathbb{Z}}\right )$ for odd primes~$p$. When $k$ has the form $p^t-1$, the result is well known, but these are exceptional cases in which the homology has polynomial growth. We find that in general the homology has exponential growth and in some cases also has higher $p$-torsion. The calculations are applied to construct a $p$-local product decomposition of $ \Omega J_{k}\left (S^{2n}\right )$ for $k

The aim of this work is to construct a cohomology theory controlling the deformations of a general Drinfel\textprime d algebra $A$ and thus finish the program which began in~\cite{MS}, \cite{shnider-sternberg:preprint}. The task is accomplished in three steps. The first step, which was taken in the aforementioned articles, is the construction of a modified cobar complex adapted to a non-coassociative comultiplication. The following two steps each involves a new, highly non-trivial, construction. The first construction, essentially combinatorial, defines a differential graded Lie algebra structure on the simplicial chain complex of the associahedra. The second construction, of a more algebraic nature, is the definition of a map of differential graded Lie algebras from the complex defined above to the algebra of derivations on the bar resolution. Using the existence of this map and the acyclicity of the associahedra we can define a so-called homotopy comodule structure (Definition~3.3 below) on the bar resolution of a general Drinfel\textprime d algebra. This in turn allows us to define the desired cohomology theory in terms of a complex which consists, roughly speaking, of the bimodule and bicomodule maps from the bar resolution to the modified cobar resolution. The complex is bigraded but not a bicomplex as in the Gerstenhaber-Schack theory for bialgebra deformations. The new components of the coboundary operator are defined via the constructions mentioned above. The results of the paper were announced in [12]. [ABSTRACT FROM AUTHOR]

In this paper we study compact submanifolds of Euclidean space with nonnegative isotropic curvature and low codimension. We determine their homology completely in the case of hypersurfaces and for some low codimensional conformally flat immersions. [ABSTRACT FROM AUTHOR]