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Questions tagged [betti-numbers]

This tag is for questions about Betti numbers. In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes.

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How to compute the Second and higher order Betti numbers of a graph?

I know that the zeroth Betti number is the number of connected components of a graph, and the first one is computed using Euler characteristics. However, I am not sure if we can compute the higher ...
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1answer
127 views

Why does the Betti number give the measure of k-dimensional holes?

I was reading Paul Renteln "MANIFOLDS, TENSORS, AND FORMS An Introduction for Mathematicians and Physicists" p.145, where he defined the Betti number as $dim H_m(K)$, where $H_m(K)$ is the quotient ...
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41 views

Betti numbers of the Stanley-Reisner ring of a simplicial complex which is the cycle on $n$-vertices

Let $\Delta$ be a simplicial complex which is the cycle on $n$-vertices $V=\{x_1,...,x_n\}$ (say) i.e. the facets of $\Delta$ are $\{x_i, x_{i+1}\}$ for $1\le i\le n$ with $x_{n+1}=x_1$. Let $S=k[x_1,....
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55 views

Betti Numbers' inequality and Mayer-Vietoris sequence

Let $U, V\subset M$ be open. How the Mayer-Vietoris sequence, $$\to H^{i-1}(U \cap V) \to H^i(U \cup V)\to H^i(U)\oplus H^i(V)\to$$ leads immediately to the $$b^i(U \cup V) \leq b^i(U) + b^i(V) + b^{...
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1answer
86 views

The second Betti number of a group

First I can't find the definition of second Betti number of a group. (Can you tell me any reference about this definition?) Also I don't know why $b_2(M)\ge b_2(G)$, where $M$ is a manifold with ...
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0answers
31 views

Betti sum, cup-length, Lusternik-Schnirelmann category, and critical points

I am trying to make some order in the notions of cup-length, sum of Betti numbers, LS category, critical points of functions. Let $M$ be smooth closed compact manifold. We denote by Crit($M$) the ...
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1answer
67 views

Closed oriented manifold with middle Betti is one with odd degree.

The rational cohomology ring of complex projective plane $\mathbb{CP}^{2}$ is truncated polynomial ring $\frac{\mathbb{Q}[X]}{(X)^{3}},\,\,deg(X)=2$. In this case, the degree of a generator is 2. Is ...
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1answer
55 views

Closed oriented even dimensional manifold with only three non-zero Betti numbers.

The complex and quaternionic projective planes are the examples of a closed oriented even dimensional manifold with exactly three non-zero Betti numbers. For more example see the paper ''Rational ...
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1answer
22 views

Top non-zero Betti number of connected manifold of finite type.

The Top non-zero Betti number of a closed oriented manifold is one. is it true for the general manifold or not?
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calculate Betti numbers of a specific polynomial variety

My question is: I am interested in calculating the Betti numbers of a specific polynomial variety (w.r.t. singular cohomology) whose zeros I am looking at over $\mathbb{C}$ (it has integer ...
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73 views

Relationship between Betti numbers $b_i(M;\mathbb{Q})$ and the dimension of rational homotopy $\dim_{\mathbb{Q}}\pi_i(M)\otimes\mathbb{Q}$

What is the relationship between the Betti numbers $b_i(M;\mathbb{Q})=rkH_i(M;\mathbb{Q})$ of a Riemannian manifold $M$ and the dimension of rational homotopy $\dim_{\mathbb{Q}}\pi_i(M)\otimes\mathbb{...
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1answer
173 views

What are the Betti numbers of a double pinched torus?

What are the betti numbers for a double pinched torus? A intuitive explanation for which holes these betti numbers correspond to would also be much appreciated.
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66 views

table Betti numbers for real Grassmannians

I am looking for a table of Betti numbers for real oriented and not oriented Grassmannians. Is there some references to get this ?
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389 views

Calculating Betti numbers of torus

I'm beginning to study some algebraic topology and I'm having some trouble working out the example of the torus. I want to calculate its Betti numbers, which are described as the "$k$-dimensional ...
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48 views

Persistence Homology on a grid Distance measure

I am just beginning to learn about topological data analysis and understand the basics. With respect to constructing a persistence diagram, I understand level sets etc. My question is regarding how ...
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0answers
44 views

Betti numbers and dominant maps?

Assume that $f:X\rightarrow Y$ is dominant morphism of algebraic varieties. Is there any consequences or relations between the betti numbers or the Hodge numbers of $X$ and $Y$?
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1answer
207 views

The Betti number of complex projective spaces

I have known from the wikipedia that the Betti number of the complex projective space run 1, 0, 1, 0, ..., 0, 1, 0, 0, 0, ... That is, 0 in odd dimensions, 1 in even dimensions up to 2n.  However,I ...
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computing ζ function of some simple varieties over finite field

The weil conjectures can be proved by using some algebraic geometry methods and the establish of etale cohomology. However, can I compute ζ functions over finte field for some simple examples of ...
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68 views

Betti numbers of a group times a complex variety

I am currently studying cohomology groups of complex algebraic varieties, spending a lot of time on computing Betti numbers. I came across several manifolds which have a discrete symmetry group (like ...
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2answers
80 views

Practical question on computation of Betti numbers

Maybe my question has an answer here, but I don't understand that. I will pose mine differently. The following definitions are from [1, Sec 4.1]. Consider a simplicial complex. The $k$-th chain group ...
4
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1answer
114 views

A compact flat manifold whose first Betti number is equal to the dimension is a flat torus

I know the following to be true: If $(M,g)$ is a compact flat Riemannian manifold whose first Betti number ($= \dim H_{dR}^1(M)$) is equal to the dimension, then it is (isometric to) a flat torus. ...
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Betti numbers change after homogenization

I'm trying to understand the following sentence: "Betti numbers usually change after homogenization". I understood it as follows, and I hope you can tell me if there is another interpretation or if I ...
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226 views

Do manifolds have the homotopy type of finite-dimensional CW-complexes?

It is well know that a topological manifold $M$ is homotopy equivalent to a CW-complex $X$. In addition, if $M$ is compact, one even has $M \simeq X$ for some finite CW-complex. In a Mathoverflow ...
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1answer
311 views

Why is homology unable to distinguish between planar and non-planar graphs?

Intuitively, one can think of non-planar graphs as having a "two-dimensional hole" (Edit: this is both unclear and not entirely, if at all, correct, see comments below question), thus necessitating ...
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1answer
69 views

Steps inbetween? Weil's zeta function

Why is it that, that is "just" what the Zeta function is? What happened in between? I messed around with it for roughly an hour and couldn't get it to come out right. The second photo is just for ...
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An example of infinite Betti number refers to G-covering, and the motivation of G-covering

I saw from a book that if G is infinite, then when considering G-covering: $p:X'\to X$, the P-th Betti number of $X'$ can be infinite. Can you give me an example of that. Also I am curious why we need ...
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2answers
336 views

Betti numbers of complex “sphere”

Let $X$ be the set of solutions to $x_1^2+\ldots+x_n^2=1$ in $\mathbb{C}^n$. This has real dimension $2(n-1)$, but since $X$ is an affine algebraic variety, the only possible non-zero topological ...
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1answer
174 views

Two ways to split the second Betti number

The definition of the positive and negative parts of the second Betti number which I know is via the diagonalized intersection form, and possible for $4$-manifolds $M$. $b^\pm_2:= \dim H^2_\pm( M;\...
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2answers
1k views

Euler characteristic: dependence on coefficients

Let $X$ be a finite CW complex and $\chi(X)$ its Euler characteristic (defined using integer coefficients). When is it true that $\chi(X)=\sum (-1)^i \dim H_i(X;F)$, where $F$ is a field? I thought ...
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1answer
558 views

Homology class and Betti number for a compact manifold with boundaries

If I take $Q=\mathbb{Z}_N \equiv \mathbb{Z}/(N \mathbb{Z})$, for a genus-g 2-dimensional Riemann surface $\Sigma$, I should have $$H_1(\Sigma; \mathbb{Z}_N)=\prod^{2g}_1 \mathbb{Z}_N,$$ So, $$|H_1(\...
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1answer
772 views

Question about the Betti numbers

Definition of Betti number at http://en.wikipedia.org/wiki/Betti_number The $n^{th}$ Betti number represents the rank of the $n^{th}$ homology group, denoted $H_n$ "which tells us the maximum amount ...
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1answer
354 views

About Betti Numbers

I'm studying the book 'The Geometry of Syzygies' of David Eisenbud, but I'm having problem with the following step, in page 7 he says the we have a free resolution to the set of ten points in $\mathbb{...
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1answer
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What is the Betti number of a group?

I'm studying the Fundamental Theorem of finitely generated Abelian group, and it says that the number of factors equal to $\mathbb Z$ (textbook says it is the Betti number of the group) is unique up ...
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163 views

Graded Betti Numbers of a Graded Ideal with Linear Quotients

Exercise 8.8(a) in Monomial Ideals by Herzog and Hibi: Let $I\subset S=K[x_{1},...,x_{n}]$ be a graded ideal which has linear quotients with respect to a homogeneous system of generators $f_{1},...,...
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Graded Betti Numbers of a Stable Monomial Ideal

Exercise 8.8 in Monomial Ideals by Herzog and Hibi: Let $I\subset S=K[x_{1},...,x_{n}]$ be a stable monomial ideal with $G(I)=\{u_{1},...,u_{m}\}$ and such that for $i<j$, either $\deg(u_{i})<...
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1answer
175 views

Castelnuovo-Mumford regularity and Betti numbers: an existence question

Let $k$ be a field and $M$ a finitely generated, graded module over the graded ring $S=k[x_1,\dots,x_n]$. Let $\cdots \rightarrow F_j \rightarrow F_{j-1} \rightarrow \cdots F_1 \rightarrow F_0 \...
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How does one show that definition of Betti number and its “informal definition” are equal?

When formally defining Betti number, we often use homology group - but I am not sure how we can use that definition to prove the informal definition of Betti number - that talks about "unconnected and ...
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1answer
476 views

on the definition of graded Betti numbers

Let's use as reference the slides 19-31. Let $S=k[x_1,\cdots,x_n]$ and $M$ a finitely generated graded $S$-module. Then by Hilbert's Syzygy Theorem, $M$ has a minimal, graded, free resolution of ...
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1answer
236 views

Euler number in terms of Betti numbers

This is related to this question. In the Paper On the Mordell-Weil lattices (p. 28, Lemma 10.1) it is proved that the rank $\rho$ of the Néron-Severi lattice of a rational elliptic surface is 10. In ...
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1answer
345 views

Computing Betti numbers using Macaulay2

Let $k$ be a field and $R=k[x,y,z]$, let $M=R/\langle x^2,xy,yz^2,y^4\rangle$ be $R$-module, how can we compute the left free resolution of $M$, and also the Betti numbers of this resolution?
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1answer
454 views

Betti Numbers with coefficients in reals, rationals & integers.

One knows from the Universal Coefficient Theorem that Integral Homology can be used to derive homology with coefficients in any other groups like e.g. Reals, Rationals, Z/2Z etc. Suppose you have ...
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1answer
278 views

comparing Betti numbers

My question is about what one could say about the Betti number of both spaces $X$ and $Y$ relative to one another if we have a map $f$ between them (e.g., a classical case is when $f$ is a covering ...