Cup Product Research Articles

Hodge Integrals in FJRW Theory

We study higher-genus Fan–Jarvis–Ruan–Witten theory of any chain polynomial with any group of symmetries. Precisely, we give an explicit way to compute the cup product of Polishchuk and Vaintrob’s virtual class with the top Chern class of the Hodge bundle. Our formula for this product holds in any genus and without any assumption on the semi-simplicity of the underlying cohomological field theory.

KOSZUL CALCULUS

AbstractWe present a calculus that is well-adapted to homogeneous quadratic algebras. We define this calculus on Koszul cohomology – resp. homology – by cup products – resp. cap products. The Koszul homology and cohomology are interpreted in terms of derived categories. If the algebra is not Koszul, then Koszul (co)homology provides different information than Hochschild (co)homology. As an application of our calculus, the Koszul duality for Koszul cohomology algebras is proved foranyquadratic algebra, and this duality is extended in some sense to Koszul homology. So, the true nature of the Koszul duality theorem is independent of any assumption on the quadratic algebra. We compute explicitly this calculus on a non-Koszul example.

Cup products in surface bundles, higher Johnson invariants, and MMM classes

In this paper we prove a family of results connecting the problem of computing cup products in surface bundles to various other objects that appear in the theory of the cohomology of the mapping class group $${\text {Mod}}_g$$ and the Torelli group $$\mathcal {I}_g$$ . We show that Kawazumi’s twisted MMM class $$m_{0,k}$$ can be used to compute k-fold cup products in surface bundles, and that $$m_{0,k}$$ provides an extension of the higher Johnson invariant $$\tau _{k-2}$$ to $$H^{k-2}({\text {Mod}}_{g,*}, \wedge ^k H_1)$$ . These results are used to show that the behavior of the restriction of the even MMM classes $$e_{2i}$$ to $$H^{4i}(\mathcal {I}_g^1)$$ is completely determined by $${\text {im}}(\tau _{4i}) \le \wedge ^{4i+2}H_1$$ , and to give a partial answer to a question of D. Johnson. We also use these ideas to show that all surface bundles with monodromy in the Johnson kernel $$\mathcal K_{g,*}$$ have cohomology rings isomorphic to that of a trivial bundle, implying the vanishing of all $$\tau _i$$ when restricted to $$\mathcal K_{g,*}$$ .

On 5–manifolds with free fundamental group and simple boundary links in S5

We classify compact oriented $5$-manifolds with free fundamental group and $\pi_{2}$ a torsion free abelian group in terms of the second homotopy group considered as $\pi_1$-module, the cup product on the second cohomology of the universal covering, and the second Stiefel-Whitney class of the universal covering. We apply this to the classification of simple boundary links of $3$-spheres in $S^5$. Using this we give a complete algebraic picture of closed $5$-manifolds with free fundamental group and trivial second homology group.

Comparison morphisms between two projective resolutions of monomial algebras

We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution $\operatorname{\mathbb{Bar}} A$ and Bardzell's resolution $\operatorname{\mathbb{Ap}} A$; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $\operatorname{HH} ^*(A)$ and the second one has been shown to be an efficient tool for computation of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for $A= \mathbb{k} Q/I$ a monomial algebra such that $\dim_ \mathbb{k} e_i A e_j = 1$ whenever there exists an arrow $\alpha: i \to j \in Q_1$, we describe the Lie action of the Lie algebra $\operatorname{HH}^1(A)$ on $\operatorname{HH}^{\ast} (A)$.

Comparison morphisms between two projective resolutions of monomial algebras

We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution $\operatorname{\mathbb{Bar}} A$ and Bardzell's resolution $\operatorname{\mathbb{Ap}} A$; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $\operatorname{HH} ^*(A)$ and the second one has been shown to be an efficient tool for computation of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for $A= \mathbb{k} Q/I$ a monomial algebra such that $\dim_ \mathbb{k} e_i A e_j = 1$ whenever there exists an arrow $\alpha: i \to j \in Q_1$, we describe the Lie action of the Lie algebra $\operatorname{HH}^1(A)$ on $\operatorname{HH}^{\ast} (A)$.

Type II string theory on Calabi-Yau manifolds with torsion and non-Abelian discrete gauge symmetries

We provide the first explicit example of Type IIB string theory compactification on a globally defined Calabi-Yau threefold with torsion which results in a four-dimensional effective theory with a non-Abelian discrete gauge symmetry. Our example is based on a particular Calabi-Yau manifold, the quotient of a product of three elliptic curves by a fixed point free action of {mathbb{Z}}_2times {mathbb{Z}}_2 . Its cohomology contains torsion classes in various degrees. The main technical novelty is in determining the multiplicative structure of the (torsion part of) the cohomology ring, and in particular showing that the cup product of second cohomology torsion elements goes non-trivially to the fourth cohomology. This specifies a non-Abelian, Heisenberg-type discrete symmetry group of the cfour-dimensional theory.

Cup products of line bundles on homogeneous varieties and generalized PRV components of multiplicity one

Let X=G/B be a complete flag variety, and L' and L" two line bundles on X. Consider the cup product map H^{d'}(X,L') x H^{d"}(X, L") --> H^{d}(X,L), where L=L' x L" and d=d'+d". We answer two natural questions about the map above: When is it a nonzero map of irreducible G-representations? Conversely, given generic irreducible representations V' and V" of G, which irreducible components of V' x V" may appear in the right hand side of the map above? We also give bounds on the multiplicities appearing in a tensor product, and relate these considerations to the boundary of the Littlewood-Richardson cone.

Review: Utilization of Waste From Coffee Production

Abstract Coffee is one of the most valuable primary products in the world trade, and also a central and popular part of our culture. However, coffees production generate a lot of coffee wastes and by-products, which, on the one hand, could be used for more applications (sorbent for the removal of heavy metals and dyes from aqueous solutions, production of fuel pellets or briquettes, substrate for biogas, bioethanol or biodiesel production, composting material, production of reusable cups, substrat for mushroom production, source of natural phenolic antioxidants etc.), but, on the other hand, it could be a source of severe contamination posing a serious environmental problem. In this paper, we present an overview of utilising the waste from coffee production.

Gerstenhaber Algebra Structure on the Hochschild Cohomology of Quadratic String Algebras

We describe the Gerstenhaber algebra structure on the Hochschild cohomology HH∗(A) when A is a quadratic string algebra. First we compute the Hochschild cohomology groups using Barzdell’s resolution and we describe generators of these groups. Then we construct comparison morphisms between the bar resolution and Bardzell’s resolution in order to get formulae for the cup product and the Lie bracket. We find conditions on the bound quiver associated to string algebras in order to get non-trivial structures.

On the Andreadakis conjecture restricted to the “lower-triangular” automorphism groups of free groups

In this paper, we consider a certain subgroup [Formula: see text] of the IA-automorphism group of a free group. We determine the images of the [Formula: see text]th Johnson homomorphism restricted to [Formula: see text] for any [Formula: see text] and [Formula: see text]. By using this result, we give an affirmative answer to the Andreadakis conjecture restricted for [Formula: see text]. Namely, we show that the intersection of the Andreadakis–Johnson filtration and [Formula: see text] coincides with the lower central series of [Formula: see text]. In a series of this research, we obtain additional results on the integral (co)homology groups of [Formula: see text]. In particular, we determine the first homology group, and study the cup product of first cohomologies of [Formula: see text]. Furthermore, we construct nontrivial second homology classes of [Formula: see text] by observing its generators and relators, and show that the second cohomology group is not generated by cup products of the first cohomology groups.

Multiplicativity of perverse filtration for Hilbert schemes of fibered surfaces

Let S→C be a smooth projective surface with numerically trivial canonical bundle fibered onto a curve. We prove the multiplicativity of the perverse filtration with respect to the cup product on H⁎(S[n],Q) for the natural morphism S[n]→C(n). We also prove the multiplicativity for five families of Hitchin systems obtained in a similar way and compute the perverse numbers of the Hitchin moduli spaces. We show that for small values of n the perverse numbers match the predictions of the numerical version of the de Cataldo–Hausel–Migliorini P=W conjecture and of the conjecture by Hausel, Letellier and Rodriguez-Villegas.

Isotropy index for the connected sum and the direct product of manifolds

A subspace or subgroup is isotropic under a bilinear map if the restriction of the map on it is trivial. We study maximal isotropic subspaces or subgroups under skew-symmetric maps, and in particular the isotropy index---the maximum dimension of an isotropic subspace or maximum rank of an isotropic subgroup. For a smooth closed orientable manifold $M$, we describe the geometric meaning of the isotropic subgroups of the first cohomology group with different coefficients under the cup product. We calculate the corresponding isotropy index, as well as the set of ranks of all maximal isotropic subgroups, for the connected sum and the direct product of manifolds. Finally, we study the relationship of the isotropy index with the first Betti number and the co-rank of the fundamental group. We also discuss applications of these results to the topology of foliations.

On products in a real moment-angle manifold

In this paper we give a necessary and sufficient condition for a (real) moment-angle complex to be a topological manifold. The cup and cap products in a real moment-angle manifold are studied. Consequently, the cohomology ring (with coefficients integers) of a polyhedral product by pairs of disks and their bounding spheres is isomorphic to that of a differential graded algebra associated to $K$ and the dimensions of the disks.

Hopf ring structure on the mod p cohomology of symmetric groups

We describe a Hopf ring structure on ⊕ n≥0H∗(Σn; ℤp), discovered by Strickland and Turner, where Σn is the symmetric group of n objects and p is an odd prime. We also describe an additive basis on which the cup product is explicitly determined, compute the restriction to modular invariants and determine the action of the Steenrod algebra on our Hopf ring generators. For p = 2 this was achieved in work of Giusti, Salvatore and Sinha, of which this work is an extension.

Twisted cohomology pairings of knots I; diagrammatic computation

We provide a diagrammatic computation for the bilinear form, which is defined as the pairing between the (relative) cup products with every local coefficients and every integral homology 2-class of every link in the 3-sphere. As a corollary, we construct bilinear forms on the twisted Alexander modules of links.

A combinatorial algorithm for computing higher order linking numbers

We develop the intersection theory at relative chain-cochain level, and apply it along with the use of Seifert disks for an oriented link to give a combinatorial algorithm to compute Massey's higher order linking numbers. It is subtle to compute higher-order linking numbers, and it has been a folklore to use the intersection theory in the process, which was first suggested by W. Massey. Massey introduced the higher-order linking as an application of his higher-order cohomology operations defined in terms of suitable cochains, and he calculated the third-order linking numbers for some 3-component links by shifting from cohomology and cup product to homology and intersection theory via duality theorems for manifolds. Later several works along this direction gave methods in various forms for computing the higher-order linking numbers, and others computed Milnor's invariants by using their well-known connection to Massey invariants; but a formal derivation of the general formulae in intersection theory which are used for the computation has not yet been given, and a concrete algorithm for the computation is hence lacking. In this paper we complete Massey's original approach by developing systematically the intersection theory at the relative chain-cochain level in the simplicial category, and use it to derive recursive combinatorial formulae for computing all higher-order linking numbers.

Triple Massey products and Galois theory

We show that any triple Massey product with respect to prime 2 contains 0 whenever it is defined over any field. This extends the theorem of M. J. Hopkins and K. G. Wickelgren, from global fields to any fields. This is the first time when the vanishing of any n -Massey product for some prime p has been established for all fields. This leads to a strong restriction on the shape of relations in the maximal pro-2-quotients of absolute Galois groups, which was out of reach until now. We also develop an extension of Serre's transgression method to detect triple commutators in relations of pro- p -groups, where we do not require that all cup products vanish. We prove that all n -Massey products, n \geq 3 , vanish for general Demushkin groups. We formulate and provide evidence for two conjectures related to the structure of absolute Galois groups of fields. In each case when these conjectures can be verified, they have some interesting concrete Galois theoretic consequences. They are also related to the Bloch–Kato conjecture.

On the Hochschild cohomology ring of integral cyclic algebras

We determine the ring structure of the Hochschild cohomology HH*(Γ) of an integral cyclic algebra Γ by giving a projective bimodule resolution of Γ and calculating cup product by means of a diagonal approximation map.

Invariance of quantum rings under ordinary flops I: Quantum corrections and reduction to local models

This is the first of a sequence of papers proving the quantum invariance under ordinary flops over an arbitrary smooth base. In this first part, we determine the defect of the cup product under the canonical correspondence and show that it is corrected by the small quantum product attached to the extremal ray. We then perform various reductions to reduce the problem to the local models. In Part II, we develop a quantum Leray--Hirsch theorem and use it to show that the big quantum cohomology ring is invariant under analytic continuations in the K\ahler moduli space for ordinary flops of splitting type. In Part III, together with F. Qu, we remove the splitting condition by developing a quantum splitting principle, and hence solve the problem completely.