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1.
Polygon spaces such as , or the three‐dimensional analogs N? play an important rle in geometry and topology, and are also of interest in robotics where the li model the lengths of robot arms. When n is large, one can assume that each li is a positive real valued random variable, leading to a random manifold. The complexity of such manifolds can be approached by computing Betti numbers, the Euler characteristics, or the related Poincaré polynomial. We study the average values of Betti numbers of dimension pn when pn → ∞ as n → ∞. We also focus on the limiting mean Poincaré polynomial, in two and three dimensions. We show that in two dimensions, the mean total Betti number behaves as the total Betti number associated with the equilateral manifold where . In three dimensions, these two quantities are not any more asymptotically equivalent. We also provide asymptotics for the Poincaré polynomials. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010 相似文献
2.
W. Lück 《Geometric And Functional Analysis》2013,23(2):622-663
In this paper we consider the asymptotic behavior of invariants such as Betti numbers, minimal numbers of generators of singular homology, the order of the torsion subgroup of singular homology, and torsion invariants. We will show that all these vanish in the limit if the CW-complex under consideration fibers in a specific way. In particular we will show that all these vanish in the limit if one considers an aspherical closed manifold which admits a non-trivial S 1-action or whose fundamental group contains an infinite normal elementary amenable subgroup. By considering classifying spaces we also get results for groups. 相似文献
3.
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. 相似文献
4.
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.
Asymptotique des nombres de Betti, invariants $l^2$ et laminations相似文献
5.
Lucia Marino 《Rendiconti del Circolo Matematico di Palermo》2003,52(2):263-280
IfX is a set of distinct points in ℙ2 with given graded Betti numbers, we produce a new set of pointsY with the same graded Betti numbers asX which admits all possible conductor degrees according to the graded Betti numbers. Moreover, for such schemes we can compute
the conductor degree for each point.
We conclude by generalizing the construction of these schemes, obtaining again the same results. 相似文献
6.
Miklos Abert Nicolas Bergeron Ian Biringer Tsachik Gelander Nikolay Nikolov Jean Raimbault Iddo Samet 《Comptes Rendus Mathematique》2011,349(15-16):831-835
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. 相似文献
7.
Hannes Pouseele 《Geometriae Dedicata》2006,122(1):77-88
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. 相似文献
8.
S. Basu 《Discrete and Computational Geometry》1999,22(1):1-18
In this paper we prove new bounds on the sum of the Betti numbers of closed semi-algebraic sets and also give the first single
exponential time algorithm for computing the Euler characteristic of arbitrary closed semi-algebraic sets.
Given a closed semi-algebraic set S
R
k
defined as the intersection of a real variety, Q=0, deg(Q)≤d, whose real dimension is k', with a set defined by a quantifier-free Boolean formula with no negations with atoms of the form P
i
=0, P
i
≥ 0, P
i
≤ 0, deg(P
i
) ≤ d, 1≤ i≤ s, we prove that the sum of the Betti numbers of S is bounded by s
k'
(O(d))
k
. This result generalizes the Oleinik—Petrovsky—Thom—Milnor bound in two directions. Firstly, our bound applies to arbitrary
unions of basic closed semi-algebraic sets, not just for basic semi-algebraic sets. Secondly, the combinatorial part (the
part depending on s ) in our bound, depends on the dimension of the variety rather than that of the ambient space. It also generalizes the result
in [4] where a similar bound is proven for the number of connected components. We also prove that the sum of the Betti numbers
of S is bounded by s
k'
2
O(k2 m4)
in case the total number of monomials occurring in the polynomials in is m. Using the tools developed for the above results, as well as some additional techniques, we give the first single exponential
time algorithm for computing the Euler characteristic of arbitrary closed semi-algebraic sets.
Received September 9, 1997, and in revised form March 18, 1998, and October 5, 1998. 相似文献
9.
We control the number of critical points of a height function arising from the Nash isometric embedding of a compact Riemanniann-manifoldM. The Ln/2 curvature norm ∥R∥ and a similar scalar ∥R∥ are introduced and their integralR(M) andR(M) overM. We prove thatR(M) is bounded below by a constant depending only onn and the Betti numbers ofM. Thus a new sphere theorem is proved by eliminating allith Betti numbers fori = 1, .…n −1. The emphasis is that our sphere theorem imposes no restriction on the range of curvature.
Research partially supported by Grant-in-Aid for General Scientific Research, grant no. 07454018. 相似文献
10.
Marina Logares 《Geometriae Dedicata》2006,123(1):187-200
Let X be a compact Riemann surface together with a finite set of marked points. We use Morse theoretic techniques to compute the
Betti numbers of the parabolic U(2,1)-Higgs bundles moduli spaces over X. We give examples for one marked point showing that the Poincaré polynomials depend on the system of weights of the parabolic
bundle.
相似文献
11.
Jens Piontkowski 《Mathematische Zeitschrift》2007,255(1):195-226
We compute the Euler number of the compactified Jacobian of a curve whose minimal unibranched normalization has only plane irreducible singularities with characteristic Puiseux exponents (p, q), (4, 2q, s), (6, 8, s), or (6, 10, s). Further, we derive a combinatorial method to compute the Betti numbers of the compactified Jacobian of an unibranched rational curve with singularities like above. Some of the Betti numbers can be stated explicitly. 相似文献
12.
I. Yu. Limonchenko 《Mathematical Notes》2013,94(3-4):351-363
The bigraded Betti numbers β ?i,2j (P) of a simple polytope P are the dimensions of the bigraded components of the Tor groups of the face ring k[P]. The numbers β ?i,2j (P) reflect the combinatorial structure of P, as well as the topological structure of the corresponding moment-angle manifold Z P ; thus, they find numerous applications in combinatorial commutative algebra and toric topology. We calculate certain bigraded Betti numbers of the type β ?i,2(i+1) for associahedra and apply the calculation of bigraded Betti numbers for truncation polytopes to study the topology of their moment-angle manifolds. Presumably, for these two series of simple polytopes, the numbers β ?i,2j (P) attain their minimum and maximum values among all simple polytopes P of fixed dimension with a given number of facets. 相似文献
13.
Basu 《Discrete and Computational Geometry》2008,30(1):65-85
Abstract. A classic result in real algebraic geometry due to Oleinik—Petrovskii, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semi-algebraic sets. This bound is tight as one can construct examples having that
many connected components. However, till now no significantly better bounds were known on the individual higher Betti numbers.
We prove better bounds on the individual Betti numbers of basic semi-algebraic sets, as well as arrangements of algebraic
hypersurfaces. As a corollary we obtain a polynomial bound on the highest Betti numbers of basic semi-algebraic sets defined
by quadratic inequalities. 相似文献
14.
P 《Journal of Algebra》2007,318(2):619-640
The graded Betti numbers of the minimal free resolution (and also therefore the Hilbert function) of the ideal of a fat point subscheme Z of P2 are determined whenever Z is supported at any 6 or fewer distinct points. All results hold over an algebraically closed field k of arbitrary characteristic. 相似文献
15.
A. Lerario 《Discrete and Computational Geometry》2012,48(4):1025-1047
We study the topology of the set X of the solutions of a system of two quadratic inequalities in the real projective space ?P n (e.g. X is the intersection of two real quadrics). We give explicit formulas for its Betti numbers and for those of its double cover in the sphere S n ; we also give similar formulas for level sets of homogeneous quadratic maps to the plane. We discuss some applications of these results, especially in classical convexity theory. We prove the sharp bound b(X)??2n for the total Betti number of X; we show that for odd n this bound is attained only by a singular?X. In the nondegenerate case we also prove the bound on each specific Betti number b k (X)??2(k+2). 相似文献
16.
Basu 《Discrete and Computational Geometry》2003,30(1):65-85
Abstract. A classic result in real algebraic geometry due to Oleinik—Petrovskii, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semi-algebraic sets. This bound is tight as one can construct examples having that
many connected components. However, till now no significantly better bounds were known on the individual higher Betti numbers.
We prove better bounds on the individual Betti numbers of basic semi-algebraic sets, as well as arrangements of algebraic
hypersurfaces. As a corollary we obtain a polynomial bound on the highest Betti numbers of basic semi-algebraic sets defined
by quadratic inequalities. 相似文献
17.
In this paper, we study topology of the variety of closed planar n-gons with given side lengths . The moduli space where , encodes the shapes of all such n-gons. We describe the Betti numbers of the moduli spaces as functions of the length vector . We also find sharp upper bounds on the sum of Betti numbers of depending only on the number of links n. Our method is based on an observation of a remarkable interaction between Morse functions and involutions under the condition
that the fixed points of the involution coincide with the critical points of the Morse function.
相似文献
18.
We use Klee’s Dehn–Sommerville relations and other results on face numbers of homology manifolds without boundary to (i) prove
Kalai’s conjecture providing lower bounds on the f-vectors of an even-dimensional manifold with all but the middle Betti number vanishing, (ii) verify Kühnel’s conjecture that
gives an upper bound on the middle Betti number of a 2k-dimensional manifold in terms of k and the number of vertices, and (iii) partially prove Kühnel’s conjecture providing upper bounds on other Betti numbers of
odd- and even-dimensional manifolds. For manifolds with boundary, we derive an extension of Klee’s Dehn–Sommerville relations
and strengthen Kalai’s result on the number of their edges.
I. Novik research partially supported by Alfred P. Sloan Research Fellowship and NSF grant DMS-0500748.
E. Swartz research partially supported by NSF grant DMS-0600502. 相似文献
19.
Mukadas D. Missarov Roman G. Stepanov 《P-Adic Numbers, Ultrametric Analysis, and Applications》2011,3(2):114-128
In this article we consider two well known combinatorial optimization problems (travel-ling salesman and minimum spanning
tree), when n points are randomly distributed in a unit p-adic ball of dimension d. We investigate an asymptotic behavior of their solutions at large number of n. It was earlier found that the average lengths of the optimal solutions in both problems are of order n
1−1/d
. Here we show that standard deviations of the optimal lengths are of order n
1/2−1/d
if d > 1, and prove that large number laws are valid only for special subsequences of n. 相似文献
20.
We get results in Orlicz spaces L φ about best local approximation on non-balanced neighborhoods when φ satisfies a certain asymptotic condition. This fact generalizes known previous results in L p spaces. 相似文献