Homotopy invariance of Novikov-Shubin invariants and Betti numbers

We give short proofs of the Gromov-Shubin theorem on the homotopy invariance of the Novikov-Shubin invariants and of the Dodziuk theorem on the homotopy invariance of the Betti numbers of the universal covering of a closed manifold in this paper. We show that the homotopy invariance of these invariants is no more difficult to prove than the homotopy invariance of ordinary homology theory.

     

Maximal Betti numbers

We provide a short proof that the lexicographic ideal has the greatest Betti numbers among all graded ideals with a fixed Hilbert function.

     

Homotopy invariance of AF-embeddability

We prove that AF-embeddability is a homotopy invariant in the class of separable exact C ^*-algebras. This work was inspired by Spielberg's work on homotopy invariance of AF-embeddability and Dadarlat's serial works on AF-embeddability of residually finite dimensional C ^*-algebras. Submitted: February 2002.     

Growth of Betti numbers

We discuss growth rates of Betti numbers in a family of coverings of a compact cell complex X, when the corresponding L² Betti number of X is zero. We show that the Betti numbers are bounded by a function, sub-linear in the order of the covering. If the appropriate Novikov-Shubin invariant of X is positive, the rate bounds are improved. For well behaved families (such as congruence covers of symmetric spaces), if the L² spectrum of X? has a gap at zero then the growth rate is bounded by the order of the covering raised to a power less than one.     

Betti and Tachibana numbers

The Tachibana numbers t _r (M), the Killing numbers k _r (M), and the planarity numbers p _r (M) are considered as the dimensions of the vector spaces of, respectively, all, coclosed, and closed conformal Killing r-forms with 1 ≤ r ≤ n ? 1 “globally” defined on a compact Riemannian n-manifold (M,g), n >- 2. Their relationship with the Betti numbers b _r (M) is investigated. In particular, it is proved that if b _r (M) = 0, then the corresponding Tachibana number has the form t _r (M) = k _r (M) + p _r (M) for t _r (M) > k _r (M) > 0. In the special case where b ₁(M) = 0 and t ₁(M) > k ₁(M) > 0, the manifold (M,g) is conformally diffeomorphic to the Euclidean sphere.     

Betti numbers of determinantal ideals

Let R=k[x₁,…,x_n] be a polynomial ring and let IR be a graded ideal. In [T. Römer, Betti numbers and shifts in minimal graded free resolutions, arXiv: AC/070119], Römer asked whether under the Cohen–Macaulay assumption the ith Betti number β_i(R/I) can be bounded above by a function of the maximal shifts in the minimal graded free R-resolution of R/I as well as bounded below by a function of the minimal shifts. The goal of this paper is to establish such bounds for graded Cohen–Macaulay algebras k[x₁,…,x_n]/I when I is a standard determinantal ideal of arbitrary codimension. We also discuss other examples as well as when these bounds are sharp.     

Betti numbers of congruence groups

It is shown that, for certain congruence families of Galois coverings of a manifold, the individual Betti numbers are polynomial periodic functions of the level. Similar results are proved for the dimensions of other spaces of automorphic forms.     

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)     

Consecutive cancellations in Betti numbers

Let be a homogeneous ideal in a polynomial ring over a field. By Macaulay's Theorem, there exists a lexicographic ideal with the same Hilbert function as . We prove that the graded Betti numbers of are obtained from those of by a sequence of consecutive cancellations.

     

Homotopy invariance of coherent Witt groups

 We prove the following general homotopy invariance theorem for coherent Witt groups: Let be a flat morphism of separated Gorenstein schemes of finite Krull dimension with affine fibers,i.e. π⁻¹(y) is an affine space over the residue field k(y) for all yY, then the induced homomorphism of coherent Witt groups is an isomorphism. As an application we calculate the (classical) Witt group of the affine hyperbolic sphere over a regular local ring. Received: 22 August 2001; in final form: 22 June 2002 / Published online: 1 April 2003     

Betti numbers of some monomial ideals

In this paper we compute the graded Betti numbers of certain monomial ideals that are not stable. As a consequence we prove a conjecture, stated by G. Fatabbi, on the graded Betti numbers of two general fat points in

     

Betti numbers of mixed product ideals

We compute the Betti numbers of the resolution of a special class of square-free monomial ideals, the ideals of mixed products. Moreover when these ideals are Cohen-Macaulay we calculate their type. Received: 9 March 2008     

Multigraded Betti numbers of simplicial forests

We prove that multigraded Betti numbers of a simplicial forest are always either 0 or 1. Moreover a nonzero multidegree appears exactly in one homological degree in the resolution. Our work generalizes work of Bouchat [2] on edge ideals of graph trees.     

Extremal Betti numbers of edge ideals

Archiv der Mathematik - Let A be an algebra over a field F of characteristic zero. For every $$n\ge 1$$ , let $$\delta _n(A)$$ be the number of linearly independent multilinear proper central...     

Extremal Betti numbers of graded ideals

The possible extremal Betti numbers of graded ideals in the polynomial ring K[x₁,…,x_n] in n variables with coefficients in a field K are studied, completing our results in [7]. In case char(K) = 0 we determine, given any integers r < n, the conditions under which there exists a graded ideal I ? K[x₁,…, x_n] with extremal Betti numbers $\beta_{k_{i}k_{i}+\ell_{i}}\ {\rm for}\ i=1,\cdots,r$ . We also treat a similar problem for squarefree lexsegment ideals.     

Onf-vectors and Betti numbers of multicomplexes

A multicomplexM is a collection of monomials closed under divisibility. For suchM we construct a cell complex M whosei-dimensional cells are in bijection with thef _i monomials ofM of degreei+1. The bijection is such that the inclusion relation of cells corresponds to divisibility of monomials. We then study relations between the numbersf _i and the Betti numbers of M. For squarefree monomials the construction specializes to the standard geometric realization of a simplicial complex.This work was supported by the Mittag-Leffler Institute during the Combinatorial Year program 1991–92. The second author also acknowledges support from the Serbian Science Foundation, Grant No. 0401D.     

Betti numbers of 3-Sasakian manifolds

A vanishing theorem and constraints are given for the Betti numbers of compact 3-Sasakian manifolds.     

Betti numbers of perfect homogeneous ideals

In this paper we study the graded minimal free resolution of the ideal, I, of any arithmetically Cohen-Macaulay projective variety. First we determine the range of the shifts (twisting numbers) that can possibly occur in the resolution, in terms of the Hilbert function of I. Then we find conditions under which some of the twisting numbers do not occur. Finally, in some ‘good’ cases, all the Betti numbers are (recursively) computed, in terms of the Hilbert function of I or that of Extⁿ_R(R/I,R), where R is a polynomial ring over a field and n is the height of I in R.