All Questions
Tagged with euler-characteristics at.algebraic-topology
27 questions
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Noether's formula for real algebraic surfaces
Is there a version of Noether's formula for the Euler characteristic of a surface for Real algebraic surfaces?
Specifically, given $X$ a real algebraic compact smooth surface, what is the relationship ...
4
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0
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202
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Possible Euler characteristics of manifolds with tangential structures
Let $p:B\to BO$ be a fibration. We say that a manifold has a $B$-structure if its stable tangent bundle lifts to $B$. I am interested in the question of whether there exists, for a given even ...
4
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0
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202
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Fibrations and Euler characteristics with bad fundamental group
Consider a fibration $F\to E\to B$ where $H^i(F;\mathbb{Q})$ and $H^i(B;\mathbb{Q})$ are finite-dimensional, and they vanish for $i\gg 0$, and $B$ is connected. However, we do not assume that $B$ is ...
- 56.9k
13
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2
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586
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When are bundles of odd and even differential forms isomorphic?
Let $M$ be a compact oriented $n$-manifold. Denote $\Omega^k := {\bigwedge}^k T^*M$ the vector bundle of differential $k$-forms, and let $\Omega^{\text{odd}} := \bigoplus_{\text{$k$ odd}} \Omega^k$ ...
- 18.2k
5
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244
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Does the (Poincare) dual complex represent the same topology?
To start with, consider some abstract $3$-dimensional simplicial complex $\Delta$ representing a manifold without boundary, for simplicity. Then, there is this well-known construction of the "(...
- 1,171
3
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1
answer
539
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Euler characteristic of pseudomanifolds with boundary
It is a well-known fact that for every compact oriented odd-dimensional manifold $\mathcal{M}$ with boundary it holds that
$$\chi(\mathcal{M})=\frac{1}{2}\chi(\partial\mathcal{M}).$$
In particular, if ...
- 12.3k
7
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181
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In what sense do the real and complex places correspond to setting q equal to 1 or -1?
It often happens that if we have a scheme $X/\mathbb Z$ (or an open subset thereof) and we denote by $p(q) = X(\mathbb F_q)$, then $p(1)$ and $p(-1)$ compute the euler characteristic of $X(\mathbb C)$ ...
- 7,746
8
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2
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294
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Euler characteristic of the simplicial complex of sets of elements in a semilattice with non-zero meet
In a combinatorial computation, I came across the following quantity:
Consider a finite meet semilattice $L$, that is, a finite poset which is closed under $\min$. Denote the least element of $L$ by $...
- 50.8k
2
votes
0
answers
98
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Compact $G$-ENR's and Euler characteristic computed with Alexander-Spanier cohomology with compact support
Let $(Z,A)$ a compact ENR pair, then
$$\chi(Z)=\chi_c(Z-A)+\chi(A)$$
where $\chi_c$ is the Euler characteristic taken in Alexander-Spanier cohomology with compact support (ENR means euclidean ...
- 795
3
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126
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Properties of a generalization (regularization) of the Euler characteristic?
Intro: This question is about a version of the Euler characteristic for infinite dimensional chain complexes. I have no idea if this is a pre-existing concept, that's essentially what my query is ...
CommunityBot
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427
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On Lefschetz theorem and sum of Betti numbers as lower bounds for fixed points
Let $M$ be a closed manifold with holomorphic cell decomposition (if it is complex), or at least with only even cohomology. In particular, its Euler characteristic is equal to the sum of its Betti ...
CommunityBot
- 1
16
votes
3
answers
3k
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Multiplicativity of Euler characteristic for non-orientable fibrations
Let $E\to B$ be a fibration with fiber F, and assume for simplicity that B is connected. Suppose moreover that B and F have Euler characteristics (perhaps they are manifolds). Then often, one can ...
- 66.8k
7
votes
1
answer
615
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Compactification of open manifolds in the form of a manifold( with zero Euler characteristic)
Edit: According to the interesting comments of Michael Albanese and Nick L we revise the question as follows:
By manifold compactification of a manifold $M$ we mean a compact manifold $\tilde{M}$ ...
- 6,995
3
votes
1
answer
688
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Higher Euler characteristics (possible generalizations)
Let $X$ be projective and Gorenstein (over $\mathbb{C}$), of dimension $n$, then $\chi(\mathcal{O}_X)=(-1)^n\chi(\omega_X)$. Hence a "generalization": $\chi(\omega^{\otimes k}_X)$.
I'd like ...
- 23.4k
9
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1
answer
561
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"Mathai-Quillen-type" form on $M\times M$?
Let $(M,g)$ be a compact, oriented, $(2n)$-dimensional Riemannian manifold. I'm wondering whether there is a "canonical" construction of a $(2n)$-form $\eta_g$ on $M\times M$, such that
$\eta_g$ is ...
CommunityBot
- 1
10
votes
2
answers
2k
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For which classes of topological spaces Euler characteristics is defined?
I would like to know something more than what is written on wikipedia http://en.wikipedia.org/wiki/Euler_characteristic
What would be some large (largest?) class of topological spaces for which $\chi$...
- 909
1
vote
0
answers
497
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Invariance of combinatorial/geometric euler characteristic
I am trying to read and understand the paper:
TARGET ENUMERATION VIA EULER CHARACTERISTIC INTEGRALS
by YULIY BARYSHNIKOV AND
ROBERT GHRIST.
And I am having trouble with a statement. First of all, ...
CommunityBot
- 1
10
votes
2
answers
1k
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Euler characteristic, Gauss-Bonnet, and a product formula
I know very little about the Pfaffian or how it works, and I'm new at Riemannian geometry in general. But I was wondering if there is some way to make this "intuitive" argument for the fact that a ...
- 366
5
votes
1
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690
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What is the Euler characteristic of a mapping space?
Suppose that $A$ and $B$ are topological spaces homotopy equivalent to finite cell complexes, and let $B^A = \mathrm{maps}(A,B)$ denote the space of maps from $A$ to $B$. Is it there a formula for ...
- 118k
6
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0
answers
289
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Is Euler-characteristic of a simplicial complex on $n$ vertices and $f$ facets at most $n^{O(\log f)}$?
(Definition: Facet = Maximal Face)
This question is a continuation of the previous one that I had asked a couple of years ago: Is Euler characteristic of a simplicial complex upper bounded by a ...
CommunityBot
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2
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answers
1k
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What kinds of manifolds admit non-vanishing vector fields defining convergent congruences?
One of the corollaries of the Poincaré–Hopf index theorem is that a closed, connected manifold $M$ admits non-vanishing vector fields iff its Euler characteristic is zero; i.e. $\chi(M) = 0$.
I am ...
27
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2
answers
3k
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Euler Characteristic of a manifold with non-vanishing vector field,
A friend of mine recently asked me if I knew any simple, conceptual argument (even one that is perhaps only heuristic) to show that if a triangulated manifold has a non-vanishing vector field, then ...
- 26.3k
10
votes
2
answers
703
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When does an even-dimensional manifold fiber over an odd-dimensional manifold?
Are there simple necessary and sufficient conditions for an (oriented) even-dimensional compact smooth manifold to fiber over an (oriented) odd-dimensional manifold (with oriented fibers)?
For ...
- 18.9k
11
votes
1
answer
968
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Chern numbers via Euler characteristics?
Let $X$ be a space good enough to have a fundamental class, and $E$ a complex vector bundle on $X$. Let $P$ be some polynomial expression, and say I want to evaluate $P(c_i(E)) \cap [X]$.
Is ...
- 27.9k
7
votes
2
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676
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Is the Euler characteristic of aspherical connected 2-complexes at most 1? (No!) What can be said about subcomplexes of 2-complexes deformation retractible onto graphs.
I have several related questions, i do not know which one is more important to me, i think it would depend on their answers.
Is it true that the Euler characteristic of a finite connected aspherical ...
- 3,908
4
votes
0
answers
314
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Combining Lefschetz numbers with Euler classes
Given an $n$-manifold $M$ (say), we can talk about its Euler characteristic
$\chi(M)$.
This can be generalized to the Euler number of any $n$-dimensional
bundle ${\mathcal V}$. Or indeed, the Euler ...
- 27.9k
20
votes
2
answers
1k
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What's the cell structure of K(Z/nZ, 1)? Does it let me sum this divergent series? What about other finite groups?
The Eilenberg-Maclane space $K(\mathbb{Z}/2\mathbb{Z}, 1)$ has a particularly simple cell structure: it has exactly one cell of each dimension. This means that its "Euler characteristic" should be ...
- 27.7k