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.
Questions tagged [betti-numbers]
77 questions
14
votes
2 answers
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 it would be true for all fields, but I noticed that…
Angry Euler
- 161
9
votes
1 answer
On the definition of graded Betti numbers
Let's use as reference the slides 19-31.
Let $S=k[x_1,\dots,x_n]$ and $M$ a finitely generated graded $S$-module. Then by Hilbert's Syzygy Theorem, $M$ has a minimal, graded, free resolution of length at most $n$, i.e.,
$$0 \rightarrow F_m…
Manos
- 26,949
7
votes
1 answer
What is the Betti number of a group?
I'm studying the Fundamental Theorem of finitely generated Abelian group, and it says that the number of factors equal to $\mathbb Z$ (textbook says it is the Betti number of the group) is unique up to isomorphism.
So what is "the number of…
Math-Nerd
- 219
7
votes
1 answer
Poincaré's take on Poincaré duality before the advent of cohomology?
Poincaré duality was first stated, without proof, by Henri Poincaré in 1893. It was stated in terms of Betti numbers: The $k$th and $n-k$th Betti numbers, $b_k$ and $b_{n-k}$ of a closed orientable n-manifold are equal.
$$b_k = b_{n-k}.$$
From…
wonderich
- 6,059
7
votes
1 answer
Hodge numbers of singular varieties
I just realized that Hodge numbers can be defined for every $\mathbb C$-variety, not only the smooth proper ones. At least we can define them using the Grothendieck ring $K_0(\text{Var}/\mathbb C)$ since it is generated by smooth projective…
Akatsuki
- 3,390
7
votes
1 answer
Why does the Betti number give the measure of k-dimensional holes?
I was reading Paul Renteln "MANIFOLDS, TENSORS, AND FORMS An Introduction for Mathematicians and Physicists" p.145, where he defined the Betti number as $dim H_m(K)$, where $H_m(K)$ is the quotient space of cycles modulo boundary $Z_m(K)/B_m(K)$. He…
Joe Martin
- 348
7
votes
2 answers
Betti numbers of complex "sphere"
Let $X$ be the set of solutions to $x_1^2+\ldots+x_n^2=1$ in $\mathbb{C}^n$. This has real dimension $2(n-1)$, but since $X$ is an affine algebraic variety, the only possible non-zero topological Betti numbers of $X$ are $b_0,\ldots,b_{n-1}$.
What…
User
- 792
6
votes
1 answer
Question about the Betti numbers
Definition of Betti number at http://en.wikipedia.org/wiki/Betti_number
The $n^{th}$ Betti number represents the rank of the $n^{th}$ homology group, denoted $H_n$ "which tells us the maximum amount of cuts that can be made before separating a…
Vrouvrou
- 5,233
5
votes
0 answers
Closed oriented manifold with only nonzero Betti number one?
For which $n$ there exists a closed oriented manifold $M$ of dimension $2n$ which satisfies the following Betti number condition?
$$
\beta_0=\beta_n=\beta_{2n}=1;\ \beta_i=0,\ i\ne0,n,2n.
$$
By looking at the projective planes…
cybcat
- 982
5
votes
1 answer
Betti sum, cup-length, Lusternik-Schnirelmann category, and critical points
I am trying to make some order in the notions of cup-length, sum of Betti numbers, LS category, critical points of functions.
Let $M$ be smooth closed compact manifold. We denote by Crit($M$) the minimal number of critical points of a smooth…
BrianT
- 471
- 2
- 7
5
votes
1 answer
calculate Betti numbers of a specific polynomial variety
My question is: I am interested in calculating the Betti numbers of a specific polynomial variety (w.r.t. singular cohomology) whose zeros I am looking at over $\mathbb{C}$ (it has integer coefficients). I know the polynomial explicitly.
Please…
Jyothi
- 99
5
votes
0 answers
Do manifolds have the homotopy type of finite-dimensional CW-complexes?
It is well know that a topological manifold $M$ is homotopy equivalent to a CW-complex $X$. In addition, if $M$ is compact, one even has $M \simeq X$ for some finite CW-complex.
In a Mathoverflow thread, it was discussed if one can take an…
abenthy
- 963
4
votes
1 answer
comparing Betti numbers
My question is about what one could say about the Betti number of both spaces $X$ and $Y$ relative to one another if we have a map $f$ between them (e.g., a classical case is when $f$ is a covering map). Is there an inequality if $f$ happens to be…
El Moro
- 1,651
4
votes
1 answer
Betti numbers of connected sum of real projective spaces
I know that $\beta_{1}(\sharp_{h}\mathbb{RP}^{2})=h-1$. Also it is clear:
$$\beta_{i}(\sharp_{h}\mathbb{RP}^{2n})=0 \mbox{ for }0
Michael jordan
- 333
- 2
- 9
4
votes
1 answer
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 there any closed oriented 2m-manifold with the…
David Silva
- 128