Betti Numbers Research Articles

On vanishing criteria of $L^{2}$-Betti numbers of groups

On vanishing criteria of $L^{2}$-Betti numbers of groups

Projective Cohen-Macaulay monomial curves and their affine charts

Abstract In this paper, we explore when the Betti numbers of the coordinate rings of a projective monomial curve and one of its affine charts are identical. Given an infinite field k and a sequence of relatively prime integers $$a_0 = 0< a_1< \cdots < a_n = d$$ a 0 = 0 < a 1 < ⋯ < a n = d , we consider the projective monomial curve $$\mathcal {C}\subset \mathbb {P}_k^{\,n}$$ C ⊂ P k n of degree d parametrically defined by $$x_i = u^{a_i}v^{d-a_i}$$ x i = u a i v d - a i for all $$i \in \{0,\ldots ,n\}$$ i ∈ { 0 , … , n } and its coordinate ring $$k[\mathcal {C}]$$ k [ C ] . The curve $$\mathcal {C}_1 \subset \mathbb {A}_k^n$$ C 1 ⊂ A k n with parametric equations $$x_i = t^{a_i}$$ x i = t a i for $$i \in \{1,\ldots ,n\}$$ i ∈ { 1 , … , n } is an affine chart of $$\mathcal {C}$$ C and we denote by $$k[\mathcal {C}_1]$$ k [ C 1 ] its coordinate ring. The main contribution of this paper is the introduction of a novel (Gröbner-free) combinatorial criterion that provides a sufficient condition for the equality of the Betti numbers of $$k[\mathcal {C}]$$ k [ C ] and $$k[\mathcal {C}_1]$$ k [ C 1 ] . Leveraging this criterion, we identify infinite families of projective curves satisfying this property. Also, we use our results to study the so-called shifted family of monomial curves, i.e., the family of curves associated to the sequences $$j+a_1< \cdots < j+a_n$$ j + a 1 < ⋯ < j + a n for different values of $$j \in \mathbb {N}$$ j ∈ N . In this context, Vu proved that for large enough values of j, one has an equality between the Betti numbers of the corresponding affine and projective curves. Using our results, we improve Vu’s upper bound for the least value of j such that this occurs.

Uniform Boundedness on Extremal Subsets in Alexandrov Spaces

In this paper, we study extremal subsets in Alexandrov spaces with dimension n, curvature ≥κ, and diameter ≤D. We show that the following three quantities are uniformly bounded above in terms of n, κ, and D: (1) the number of extremal subsets in an Alexandrov space; (2) the Betti numbers of an extremal subset; (3) the volume of an extremal subset. The proof is an application of essential coverings introduced by Yamaguchi.

Intersection Betti Numbers of the GIT Quotient of Quartic Plane Curves

Intersection Betti Numbers of the GIT Quotient of Quartic Plane Curves

Rational points on 3‐folds with nef anti‐canonical class over finite fields

AbstractWe prove that a geometrically integral smooth projective 3‐fold with nef anti‐canonical class and negative Kodaira dimension over a finite field of characteristic and cardinality has a rational point. Additionally, under the same assumptions on and , we show that a smooth projective 3‐fold with trivial canonical class and non‐zero first Betti number has a rational point. Our techniques rely on the Minimal Model Program to establish several structure results for generalized log Calabi–Yau 3‐fold pairs over perfect fields.

The non-orientable 4-genus of 11 crossing non-alternating knots

The non-orientable 4-genus of a knot [Formula: see text] in [Formula: see text] is defined to be the minimum first Betti number of a non-orientable surface [Formula: see text] smoothly embedded in [Formula: see text] so that [Formula: see text] bounds [Formula: see text]. We will survey the tools used to compute the non-orientable 4-genus, and use various techniques to calculate this invariant for non-alternating 11 crossing knots. We will also view obstructions to a knot bounding a Möbius band given by the double branched cover of [Formula: see text] branched over [Formula: see text].

On the asymptotic normality of persistent Betti numbers

Abstract Persistent Betti numbers are a major tool in persistent homology, a subfield of topological data analysis. Many tools in persistent homology rely on the properties of persistent Betti numbers considered as a two-dimensional stochastic process $ (r,s) \mapsto n^{-1/2} (\beta^{r,s}_q ( \mathcal{K}(n^{1/d} \mathcal{X}_n))-\mathbb{E}[\beta^{r,s}_q ( \mathcal{K}( n^{1/d} \mathcal{X}_n))])$ . So far, pointwise limit theorems have been established in various settings. In particular, the pointwise asymptotic normality of (persistent) Betti numbers has been established for stationary Poisson processes and binomial processes with constant intensity function in the so-called critical (or thermodynamic) regime; see Yogeshwaran et al. (Prob. Theory Relat. Fields167, 2017) and Hiraoka et al. (Ann. Appl. Prob.28, 2018). In this contribution, we derive a strong stabilization property (in the spirit of Penrose and Yukich, Ann. Appl. Prob.11, 2001) of persistent Betti numbers, and we generalize the existing results on their asymptotic normality to the multivariate case and to a broader class of underlying Poisson and binomial processes. Most importantly, we show that multivariate asymptotic normality holds for all pairs (r, s), $0\le r\le s<\infty$ , and that it is not affected by percolation effects in the underlying random geometric graph.

On Stanley–Reisner Rings with Linear Resolutions

For a graph G, Bayer–Denker–Milutinović–Rowlands–Sundaram–Xue study in [1] a new graph complex Δkt(G), namely the simplicial complex with facets that are complements to independent sets of size k in G. They are interested in topological properties such as shellability, vertex decomposability, homotopy type, and homology of these complexes. In this paper we study more algebraic properties, such as Cohen-Macaulayness, Betti numbers, and linear resolutions of the Stanley-Reisner ring of these complexes and their Alexander duals.

Comparability of the total Betti numbers of toric ideals of graphs

The total Betti numbers of the toric ideal of a simple graph are, in general, highly sensitive to any small change of the graph. In this paper, we look at some combinatorial operations that cause total Betti numbers to change in predictable ways. In particular, we focus on a procedure that preserves these invariants.

Nonnegative Ricci curvature, splitting at infinity, and first Betti number rigidity

We study the rigidity problems for open (complete and noncompact) n-manifolds with nonnegative Ricci curvature. We prove that if an asymptotic cone of M properly contains a Euclidean Rk−1, then the first Betti number of M is at most n−k; moreover, if equality holds, then M is flat. Next, we study the geometry of the orbit Γp˜, where Γ=π1(M,p) acts on the universal cover (M˜,p˜). Under a similar asymptotic condition, we prove a geometric rigidity in terms of the growth order of Γp˜. We also give the first example of a manifold M of Ric>0 and π1(M)=Z but with a varying orbit growth order.

Stable behavior of Frobenius powers over a general hypersurface

The main goal of this paper is to prove, in positive characteristic p p , stability behavior for the graded Betti numbers in the periodic tails of the minimal resolutions of Frobenius powers of the homogeneous maximal ideals for very general choices of hypersurface in three variables whose degree has the opposite parity to that of p p . We also find some of the structure of the matrix factorization giving the resolution. We achieve this by developing a method for obtaining the degrees of the generators of the defining ideal of an c \mathfrak {c} -compressed Gorenstein Artinian graded algebra from its socle degree, where c \mathfrak {c} is a Frobenius power of the homogeneous maximal ideal. As an application, we also obtain the Hilbert–Kunz function of the hypersurface ring, as well as the Castelnuovo–Mumford regularity of the quotients by Frobenius powers of the homogeneous maximal ideal.

On the topology of determinantal links

AbstractWe study sections of the generic determinantal varieties by generic hyperplanes of various codimensions , the polar multiplicities of these sections, and the cohomology of their real and complex links. Such complex links were shown to provide the basic building blocks in a bouquet decomposition for the (determinantal) smoothings of smoothable isolated determinantal singularities. The detailed vanishing topology of such singularities was still not fully understood beyond isolated complete intersections and a few further special cases. Our results now allow to compute all the cohomology with integer coefficients below the middle degree and the middle Betti number for any determinantal smoothing.

On general fiber product rings, Poincaré series and their structure

The present paper deals with the investigation of the structure of general fiber product rings R × T S , where R, S and T are local rings with common residue field. We show that the Poincaré series of any R-module over the fiber product ring R × T S is bounded by a rational function. In addition, we give a description of depth ( R × T S ) , which is an open problem in this theory. As a biproduct, using the characterization of the Betti numbers over R × T S obtained, we provide certain cases of the Cohen-Macaulayness of R × T S and, in particular, we show that R × T S is always non-regular. Some positive answers for the Buchsbaum-Eisenbud-Horrocks and Total rank conjectures over R × T S are also established.

Definite four-manifolds with exotic smooth structures

Abstract In this paper, we study smooth structures on closed, oriented four-manifolds with fundamental group Z / 2 ⁢ Z \mathbb{Z}/2\mathbb{Z} and definite intersection form. We construct infinitely many irreducible, oriented, closed, definite, smooth four-manifolds with π 1 = Z / 2 ⁢ Z \pi_{1}=\mathbb{Z}/2\mathbb{Z} and b 2 = 1 b_{2}=1 , and b 2 = 2 b_{2}=2 . As an application, we prove that, when the second Betti number b 2 b_{2} of a definite four-manifold with π 1 = Z / 2 ⁢ Z \pi_{1}=\mathbb{Z}/2\mathbb{Z} is positive and it admits a smooth structure, then it admits infinitely many smooth structures.

Betti numbers for connected sums of graded Gorenstein Artinian algebras

The connected sum construction, which takes as input Gorenstein rings and produces new Gorenstein rings, can be considered as an algebraic analogue for the topological construction having the same name. We determine the graded Betti numbers for connected sums of graded Artinian Gorenstein algebras. Along the way, we find the graded Betti numbers for fiber products of graded rings; an analogous result was obtained in the local case by Geller [Proc. Amer. Math. Soc. 150 (2022), pp. 4159–4172]. We relate the connected sum construction to the doubling construction, which also produces Gorenstein rings. Specifically, we show that, for any number of summands, a connected sum of doublings is the doubling of a fiber product ring.

Quantum topological data analysis via the estimation of the density of states

We develop a quantum topological data analysis (QTDA) protocol based on the estimation of the density of states (DOS) of the combinatorial Laplacian. Computing topological features of graphs and simplicial complexes is crucial for analyzing data sets and building explainable artificial intelligence solutions. This task becomes computationally hard for simplicial complexes with over 60 vertices and high-degree topological features due to a combinatorial scaling. We propose to approach the task by embedding underlying hypergraphs as effective quantum Hamiltonians and evaluating their density of states from the time evolution. Specifically, we compose propagators as quantum circuits using the Cartan decomposition of effective Hamiltonians and sample overlaps of time-evolved states using multifidelity protocols. Next we develop various postprocessing routines and implement a Fourier-like transform to recover the rank (and kernel) of Hamiltonians. This enables us to estimate the Betti numbers, revealing the topological features of simplicial complexes. We test our protocol on noiseless and noisy quantum simulators and run examples on IBM quantum processors. We observe the resilience of the proposed QTDA approach to real-hardware noise even in the absence of error mitigation, showing the promise for near-term device implementations and highlighting the utility of global DOS-based estimators. Published by the American Physical Society 2024

Torsion homology growth and cheap rebuilding of inner-amenable groups

We prove that virtually torsion-free, residually finite groups that are inner-amenable and non-amenable have the cheap 1-rebuilding property, a notion recently introduced by Abért, Bergeron, Frączyk and Gaboriau. As a consequence, the first \ell^{2} -Betti number with arbitrary field coefficients and log-torsion in degree 1 vanish for these groups. This extends results previously known for amenable groups to inner-amenable groups. We use a structure theorem of Tucker-Drob for inner-amenable groups showing the existence of a chain of q -normal subgroups.

Stability of L2-Invariants on Stratified Spaces

Abstract Let $\overline{M}$ be a compact smoothly stratified pseudo-manifold endowed with a wedge metric $g$. Let $\overline{M}_{\Gamma }$ be a Galois $\Gamma $-covering. Under additional assumptions on $\overline{M}$, satisfied for example by Witt pseudo-manifolds, we show that the $L^{2}$-Betti numbers and the Novikov–Shubin invariants are well defined. We then establish their invariance under a smoothly stratified codimension-preserving homotopy equivalence, thus extending results of Dodziuk, Gromov, and Shubin to these pseudo-manifolds.

Improved algebraic fibrings

We show that a virtually residually finite rationally solvable (RFRS) group $G$ of type $\mathtt {FP}_n(\mathbb {Q})$ virtually algebraically fibres with kernel of type $\mathtt {FP}_n(\mathbb {Q})$ if and only if the first $n$$\ell ^2$-Betti numbers of $G$ vanish, that is, $b_p^{(2)}(G) = 0$ for $0 \leqslant p \leqslant n$. This confirms a conjecture of Kielak. We also offer a variant of this result over other fields, in particular in positive characteristic. As an application of the main result, we show that amenable virtually RFRS groups of type $\mathtt {FP}(\mathbb {Q})$ are virtually Abelian. It then follows that if $G$ is a virtually RFRS group of type $\mathtt {FP}(\mathbb {Q})$ such that $\mathbb {Z} G$ is Noetherian, then $G$ is virtually Abelian. This confirms a conjecture of Baer for the class of virtually RFRS groups of type $\mathtt {FP}(\mathbb {Q})$, which includes (for instance) the class of virtually compact special groups.

Frobenius-Poincaré function and Hilbert-Kunz multiplicity

We generalize the notion of Hilbert-Kunz multiplicity of a graded triple ( M , R , I ) (M,R,I) in characteristic p > 0 p>0 by proving that for any complex number y y , the limit lim n → ∞ ( 1 p n ) dim ⁡ ( M ) ∑ j = − ∞ ∞ λ ( ( M I [ p n ] M ) j ) e − i y j / p n \begin{equation*} \underset {n \to \infty }{\lim }(\frac {1}{p^n})^{\operatorname {dim}(M)}\sum \limits _{j= -\infty }^{\infty }\lambda \left ( (\frac {M}{I^{[p^n]}M})_j\right )e^{-iyj/p^n} \end{equation*} exists. We prove that the limiting function in the complex variable y y is entire and name this function the Frobenius-Poincaré function. We establish various properties of Frobenius-Poincaré functions including its relation with the tight closure of the defining ideal I I ; and relate the study Frobenius-Poincaré functions to the behaviour of graded Betti numbers of R I [ p n ] \frac {R}{I^{[p^n]}} as n n varies. Our description of Frobenius-Poincaré functions in dimension one and two and other examples raises questions on the structure of Frobenius-Poincaré functions in general.