Questions tagged [euler-characteristics]
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77 questions
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Did ancient mathematicians know Euler's characteristic for convex polyhedra?
The formula $V-E+F=2$ is so simple that I can't believe that it was really Euler (or perhaps Descartes) who first observed it (I mean the formula itself in some generality, not necessarily a valid ...
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32
votes
4
answers
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Spectrum of the Grothendieck ring of varieties
Here's a problem that may ultimately require just simple algebraic-geometry skills to be solved, or perhaps it's very deep and will never be solved at all. From the comments, some literature and my ...
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27
votes
2
answers
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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 ...
- 15.3k
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votes
0
answers
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Status of the Euler characteristic in characteristic p
In the introduction to the Asterisque 82-83 volume on `Caractérisque d'Euler-Poincaré, Verdier writes:
Enfin signalons que la situation en caractéristique positive est loin
d'être aussi ...
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votes
6
answers
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Is there a topological description of combinatorial Euler characteristic?
There are a collection of definitions of "combinatorial Euler characteristic", which is different from the "homotopy Euler characteristic". I will describe a few of them and give some references, and ...
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votes
2
answers
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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 ...
- 118k
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votes
4
answers
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Spaces that are both homotopically and cohomologically finite
Is it true that every connected space with
1) just finitely many nontrivial homotopy groups, all finite,
and
2) just finitely many nontrivial rational cohomology groups, all finite rank,
is ...
- 22.3k
17
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3
answers
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Is Euler characteristic of a simplicial complex upper bounded by a polynomial in the number of its facets ?
What is the best upper bound known on the (absolute value of) the
Euler characteristic of a simplicial complex
in terms of the number of its facets ?
In particular, I am interested in proving or ...
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16
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3
answers
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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
14
votes
4
answers
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Is the Euler characteristic a birational invariant
Suppose that $X$ and $Y$ are smooth projective varieties which are birationally equivalent. I would like to have that $$\textrm{deg} \ \textrm{td}(X) = \textrm{deg} \ \textrm{td}(Y).$$ Invoking the ...
- 9,497
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2
answers
586
views
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$ ...
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12
votes
1
answer
521
views
Source of a quote by Ferdinand Rudio
I am looking for the source and context of this quote, found e.g. at St Andrews:
Only with the greatest difficulty is one able to follow the writings of any author preceding Euler, because it was ...
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12
votes
1
answer
818
views
Does the Grothendieck ring of varieties contain torsion?
Let $K_0(Var_k)$ be the abelian group generated by the isomorphism classes of varieties over the field $k$ with the relations
$$[X]=[U]+[X\setminus U]$$
for every variety $X$ and open subvariety $U$.
...
- 3,017
11
votes
3
answers
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In Gromov-Witten theory, why is the string coupling constant weighted by $2g-2$?
Let $X$ be a Calabi-Yau threefold and let us fix a homology class $\beta\in H_2(X,\mathbb Z)$, just for simplicity. The generating series of Gromov-Witten invariants of $X$ in class $\beta$, $$\mathsf ...
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2
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What is the size of the category of finite dimensional F_q vector spaces?
The size of a finite skeletal category C in the sense of Leinster is defined as follows: Label the objects of C by integers 1,2,...,n and let aij be the number of morphisms from i to j (for i and j ...
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11
votes
1
answer
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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 ...
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11
votes
1
answer
336
views
cardinality of final coalgebras in Top
Let P be a polynomial functor from Top to Top, by which I mean a functor of the form P(X) = ∐i ≥ 0 Si × Xi where the Si are finite sets, all but finitely many of which are empty. ...
- 25.2k
11
votes
0
answers
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views
Can this be interpreted as one Euler characteristic?
Let $[n]:=\{1,\cdots,n\}$. It is known that $\{\log(p) \mid p \text{ is prime }\}$ is linearly independent over $\mathbb{Q}$. For a subset $A \subset [n]$ we can consider the matrix $L(A):=(\log(x) \...
user6671
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$...
- 28.9k
10
votes
2
answers
703
views
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 ...
- 54.6k
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 ...
- 13.5k
10
votes
1
answer
3k
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Euler Characteristic of a Variety
Let $Y$ be a "nice" scheme. I am thinking projective varieties over an algebraically closed field, for now, but I am open to more general results.
In terms of singular homology (...
- 4,902
9
votes
1
answer
561
views
"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 ...
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8
votes
2
answers
294
views
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 $...
- 305
8
votes
1
answer
243
views
Compact simply-connected homogeneous symplectic manifold
I was reading a paper in which the authors use the fact that any compact simply-connected homogeneous symplectic manifold has non-zero Euler characteristic. They prove it by quoting a theorem by ...
- 369
8
votes
1
answer
483
views
Generalized Euler characteristics of non-motivic origin
By a generalized Euler characteristic $\chi$, I mean an isomorphism invariant $\chi(V)$ inside some abelian group $A$, defined for every varietiy $V$ over a field $k$, with the property that, for all ...
- 3,017
7
votes
1
answer
500
views
Refined Euler characteristic
Is there a refinement of Euler characteristic that distinguishes between the torus $S^1 \times S^1$ and the cylinder $S^1 \times [0,1]$?
(The intuition here is that $\chi$ is multiplicative, so that $...
- 19.7k
7
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3
answers
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How to construct a vector fields with isolated zeros?
The Poincare-Hopf theorem tell us that the sum of the indices of a vector field at isolated zeros on a compact, oriented manifold is the same as the Euler characteristic of the manifold. But how to ...
- 381
7
votes
2
answers
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 ...
- 1,481
7
votes
1
answer
615
views
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}$ ...
- 366
7
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0
answers
181
views
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)$ ...
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6
votes
1
answer
519
views
Are there perverse sheaves on abelian varieties with small Euler characteristic?
Let $A$ be a simple abelian variety of dimension $g$. Let $K$ be an irreducible perverse sheaf on $A$. We know that $\chi(A,K)\geq 0$. (Corollary 1.4 of Franecki and Kapranov.) How small can $\chi(A,K)...
- 149k
6
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0
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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 ...
- 197
5
votes
1
answer
690
views
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 ...
- 54.6k
5
votes
2
answers
755
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Top chern class under finite, unramified, dominant morphism
Situation: Let $\Bbbk$ be an algebraically closed field. Assume that $\pi:Y\to X$ is an finite, dominant, unramified morphism between nonsingular varieties of dimensions $n$. Let $d=\deg(\pi)$.
What ...
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5
votes
1
answer
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The Gauss-Bonnet theorem for Sheaves
Euler Characteristic of Sheaves and the Generalized Gauss-Bonnet Theorem
Consider a sheaf $\mathscr{F}$ over a site $\mathscr{(C,J)}$, where $\mathscr{C}$ is a small category with a coverage $\mathscr{...
user62675
5
votes
1
answer
797
views
Euler characteristic of local system depends only on rank?
Let $X$ be a proper variety over a finite field $k$ of characteristic $p>0$, and let $\mathcal F$ be a finite rank $\mathbb F_\ell$ local system on (the etale site of) $X$. Is it true (and, if so, ...
- 18.7k
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2
answers
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Relating Euler characteristic, intersection product, Morse theory (plus SU(2) and 3-manifolds)
Suppose we have a (closed, oriented) 3-manifold M with a Heegard surface F of genus g. Let F* denote F with a puncture. Then the space H of representations of pi_1(F*) on SU(2) is just SU(2)^2g, and ...
- 1,129
5
votes
1
answer
283
views
Euler number for base change of a K3 surface
Suppose you have a K3 surface $S$ containing a smooth rational curve $C$ and suppose you have an elliptic fibration $S \rightarrow \mathbb P^1$ that restricts to a morphism $C \rightarrow \mathbb P^1$ ...
5
votes
0
answers
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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
4
votes
3
answers
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Morse theory and Euler characteristics
Suppose we have a space M with a real-valued, differentiable function F on M. Under what conditions on F will the Euler characteristic of M be expressed as a (signed) sum of Euler characteristics of ...
- 1,129
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votes
1
answer
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Behaviour of euler characteristics in characteristic p for finite etale covers
Let $k$ be an algebraic closure of a finite field of characteristic $p$. Fix an integer $l\neq p$. For a separated $k$-scheme $X$ of finite type, we define the (compactly supported) Euler ...
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4
votes
1
answer
898
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"Wick rotation" of tropical geometry
This question is related to my earlier, even more open-ended question on tropilcalization. I will give some background and ask my question at the end.
On R, consider the family of commutative, ...
- 54.6k
4
votes
0
answers
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
answers
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
4
votes
0
answers
99
views
When can the trace on cohomology be computed as the Euler characteristic of fixed points?
In this question all groups are finite, and all spaces are nice (eg, simplicial sets).
Given a $G$ space $X$, which we assume has finitely many nonzero cohomology groups, we can compute the trace of ...
- 1,949
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0
answers
211
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Signed number of pieces in a decomposition in the Grothendieck ring of varieties
Let $X/k$ be a (geometrically integral and connected) variety over $k$ either a field of characteristic $0$ or a finite field. Let $[X] = \sum_{i\in I}[Y_i] - \sum_{j\in J}[Z_j]$ be a decomposition ...
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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
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 ...
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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 ...
- 1,429