Quandle homology and complex volume

We introduce a new homology theory of quandles, called simplicial quandle homology, which is quite different from quandle homology developed by Carter et al. We construct a homomorphism from a quandle homology group to a simplicial quandle homology group. As an application, we obtain a method for computing the complex volume of a hyperbolic link only from its diagram.     

On homotopy groups of quandle spaces and the quandle homotopy invariant of links

For a quandle X, the quandle space BX is defined, modifying the rack space of Fenn, Rourke and Sanderson (1995) [13], and the quandle homotopy invariant of links is defined in Z[π₂(BX)], modifying the rack homotopy invariant of Fenn, Rourke and Sanderson (1995) [13]. It is known that the cocycle invariants introduced in Carter et al. (2005) [3], Carter et al. (2003) [5], Carter et al. (2001) [6] can be derived from the quandle homotopy invariant.In this paper, we show that, for a finite quandle X, π₂(BX) is finitely generated, and that, for a connected finite quandle X, π₂(BX) is finite. It follows that the space spanned by cocycle invariants for a finite quandle is finitely generated. Further, we calculate π₂(BX) for some concrete quandles. From the calculation, all cocycle invariants for those quandles are concretely presented. Moreover, we show formulas of the quandle homotopy invariant for connected sum of knots and for the mirror image of links.     

Betti Number Behavior for Nilpotent Lie Algebras

We discuss three general problems concerning the cohomology of a (real or complex) nilpotent Lie algebra: first of all, determining the Betti numbers exactly; second, determining the distribution these Betti numbers follow; and finally, estimating the size of the individual cohomology spaces or the total cohomology space. We show how spectral sequence arguments can contribute to a solution in a concrete setting. For one-dimensional extensions of a Heisenberg algebra, we determine the Betti numbers exactly. We then show that some families in this class have a M-shaped Betti number distribution, and construct the first examples with an even more exotic Betti number distribution. Finally, we discuss the construction of (co)homology classes for split metabelian Lie algebras, thus proving the Toral Rank Conjecture for this class of algebras.     

The Betti numbers of some finite racks

We show that the lower bounds for Betti numbers given in (J. Pure Appl. Algebra 157 (2001) 135) are equalities for a class of racks that includes dihedral and Alexander racks. We confirm a conjecture from the same paper by defining a splitting for the short exact sequence of quandle chain complexes. We define isomorphisms between Alexander racks of certain forms, and we also list the second and third homology groups of some dihedral and Alexander quandles.     

Cohomology of Real Diagonal Subspace Arrangements via Resolutions

We express the cohomology of the complement of a real subspace arrangement of diagonal linear subspaces in terms of the Betti numbers of a minimal free resolution. This leads to formulas for the cohomology in some cases, and also to a cohomology vanishing theorem valid for all arrangements.     

On the growth of Betti numbers of locally symmetric spaces

We announce new results concerning the asymptotic behavior of the Betti numbers of higher rank locally symmetric spaces as their volumes tend to infinity. Our main theorem is a uniform version of the Lück Approximation Theorem (Lück, 1994 [10]) which is much stronger than the linear upper bounds on Betti numbers given by Gromov in Ballmann et al. (1985) [3].The basic idea is to adapt the theory of local convergence, originally introduced for sequences of graphs of bounded degree by Benjamini and Schramm, to sequences of Riemannian manifolds. Using rigidity theory we are able to show that when the volume tends to infinity, the manifolds locally converge to the universal cover in a sufficiently strong manner that allows us to derive the convergence of the normalized Betti numbers.     

The cohomology of the Heisenberg Lie algebras over fields of finite characteristic

We give explicit formulas for the cohomology of the Heisenberg Lie algebras over fields of finite characteristic. We use this to show that in characteristic two, unlike all other cases, the Betti numbers are unimodal.

     

Springer fibers and Schubert points

Springer fibers are subvarieties of the flag variety parametrized by partitions; they are central objects of study in geometric representation theory. Schubert varieties are subvarieties of the flag variety that induce a well-known basis for the cohomology of the flag variety. This paper relates these two varieties combinatorially. We prove that the Betti numbers of the Springer fiber associated to a partition with at most three rows or two columns are equal to the Betti numbers of a specific union of Schubert varieties.     

Von Neumann Betti numbers and Novikov type inequalities

In this paper we show that Novikov type inequalities for closed 1-forms hold with the von Neumann Betti numbers replacing the Novikov numbers. As a consequence we obtain a vanishing theorem for cohomology. We also prove that von Neumann Betti numbers coincide with the Novikov numbers for free abelian coverings.

     

Betti Numbers of Small Covers and Their Two-Fold Coverings

We compute the Betti numbers of two-fold coverings of small covers with some special properties; in particular, we use the results of Davis and Januszkiewicz on the cohomology of small covers. It turned out that their proof contains some gap that we describe in detail and fill in.     

The etale sheaf cohomology of a class of singular surfaces

Summary In this paper the techniques of étale sheaf cohomology are used to estimate trigonometric sums on certain singular algebraic surfaces. This involves the computation of Betti numbers, made complicated by the presence of singularities. This leads to the introduction of some new (non-singular) surfaces. Their Betti numbers are then computed and related to the original problem which is thereby solved.

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Riassunto In questo lavoro si utilizza la coomologia étale per valutare le somme trigonometriche per certe superfici algebriche singolari. Per il calcolo dei numeri di Betti necessari, reso difficile per la presenza delle singolarità, si utilizzano delle superfici ausiliari lisce i cui numeri di Betti consentono di risolvere il problema iniziale.

     

Dirichlet to Neumann operator on differential forms

We define the Dirichlet to Neumann operator on exterior differential forms for a compact Riemannian manifold with boundary and prove that the real additive cohomology structure of the manifold is determined by the DN operator. In particular, an explicit formula is obtained which expresses Betti numbers of the manifold through the DN operator. We express also the Hilbert transform through the DN map. The Hilbert transform connects boundary traces of conjugate co-closed forms.     

Some calculations of cohomology groups of finite Alexander quandles

We calculate some quandle cohomology groups; the rational cohomology groups of any finite Alexander quandles, the second cohomology groups with a finite field coefficient of any finite Alexander quandles over a finite fields, and the third cohomology groups of the finite Alexander quandles of the form .     

Socles of Buchsbaum modules, complexes and posets

The socle of a graded Buchsbaum module is studied and is related to its local cohomology modules. This algebraic result is then applied to face enumeration of Buchsbaum simplicial complexes and posets. In particular, new necessary conditions on face numbers and Betti numbers of such complexes and posets are established. These conditions are used to settle in the affirmative Kühnel's conjecture for the maximum value of the Euler characteristic of a 2k-dimensional simplicial manifold on n vertices as well as Kalai's conjecture providing a lower bound on the number of edges of a simplicial manifold in terms of its dimension, number of vertices, and the first Betti number.     

Asymptotique des nombres de Betti,invariants $l^2$ et laminations

Let K be a finite simplicial complex. We are interested in the asymptotic behavior of the Betti numbers of a sequence of finite sheeted covers of $K$, when normalized by the index of the covers. W. Lück, has proved that for regular coverings, these sequences of numbers converge to the $l^2$ Betti numbers of the associated (in general infinite) limit regular cover of K. In this article we investigate the non regular case. We show that the sequences of normalized Betti numbers still converge. But this time the good limit object is no longer the associated limit cover of K, but a lamination by simplicial complexes. We prove that the limits of sequences of normalized Betti numbers are equal to the $l^2$ Betti numbers of this lamination. Even if the associated limit cover of K is contractible, its $l^2$ Betti numbers are in general different from those of the lamination. We construct such examples. We also give a dynamical condition for these numbers to be equal. It turns out that this condition is equivalent to a former criterion due to M. Farber. We hope that our results clarify its meaning and show to which extent it is optimal. In a second part of this paper we study non free measure-preserving ergodic actions of a countable group $\Gamma$ on a standard Borel probability space. Extending group-theoretic similar results of the second author, we obtain relations between the $l^{2}$ Betti numbers of $\Gamma$ and those of the generic stabilizers. For example, if $b_1^{(2)} (\Gamma ) \neq 0$, then either almost each stabilizer is finite or almost each stabilizer has an infinite first $l^2$ Betti number.

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Asymptotique des nombres de Betti, invariants $l^2$ et laminations

     

Cohomology structure for a Poisson algebra:Ⅱ

For a Poisson algebra, we prove that the Poisson cohomology theory introduced by Flato et al.(1995)is given by a certain derived functor. We show that the(generalized) deformation quantization is equivalent to the formal deformation for Poisson algebras under certain mild conditions. Finally we construct a long exact sequence, and use it to calculate the Poisson cohomology groups via the Yoneda-extension groups of certain quasi-Poisson modules and the Lie algebra cohomology groups.     

Simplicial orders and chordality

Chordal clutters in the sense of Bigdeli et al. (J Comb Theory Ser A 145:129–149, 2017) and Morales et al. (Ann Fac Sci Toulouse Sér 6 23(4):877–891, 2014) are defined via simplicial orders. Their circuit ideal has a linear resolution, independent of the characteristic of the base field. We show that any Betti sequence of an ideal with linear resolution appears as the Betti sequence of the circuit ideal of such a chordal clutter. Associated with any simplicial order is a sequence of integers which we call the \(\lambda \)-sequence of the chordal clutter. All possible \(\lambda \)-sequences are characterized. They are intimately related to the Hilbert function of a suitable standard graded K-algebra attached to the chordal clutter. By the \(\lambda \)-sequence of a chordal clutter, we determine other numerical invariants of the circuit ideal, such as the \(\mathbf h \)-vector and the Betti numbers.     

Betti numbers of random manifolds

Betti numbers of configuration spaces of mechanical linkages (known also as polygon spaces) depend on a large number of parameters – the lengths of the bars of the linkage. Motivated by applications in topological robotics, statistical shape theory and molecular biology, we view these lengths as random variables and study asymptotic values of the average Betti numbers as the number of links n tends to infinity. We establish a surprising fact that for a reasonably ample class of sequences of probability measures the asymptotic values of the average Betti numbers are independent of the choice of the measure. The main results of the paper apply to planar linkages as well as for linkages in R ³. We also prove results about higher moments of Betti numbers. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Triple point cancelling numbers of surface links and quandle cocycle invariants

The unknotting or triple point cancelling number of a surface link F is the least number of 1-handles for F such that the 2-knot obtained from F by surgery along them is unknotted or pseudo-ribbon, respectively. These numbers have been often studied by knot groups and Alexander invariants. On the other hand, quandle colorings and quandle cocycle invariants of surface links were introduced and applied to other aspects, including non-invertibility and triple point numbers. In this paper, we give lower bounds of the unknotting or triple point cancelling numbers of surface links by using quandle colorings and quandle cocycle invariants.