The euler-characteristics tag has no wiki summary.

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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 ...

**19**

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1k views

### 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 ...

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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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925 views

### 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 ...

**14**

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903 views

### 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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350 views

### 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 ...

**12**

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912 views

### 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 ...

**12**

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**1**answer

371 views

### 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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### 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 ...

**10**

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458 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 ...

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751 views

### 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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**1**answer

307 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. ...

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821 views

### 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 ...

**7**

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1k views

### 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 ...

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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 ...

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**1**answer

430 views

### 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 ...

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127 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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583 views

### 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 ...

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**2**answers

395 views

### 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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516 views

### 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 ...

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881 views

### 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 ...

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122 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 ...

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124 views

### 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 ...

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802 views

### 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 ...

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**1**answer

487 views

### 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 ...

**3**

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**1**answer

225 views

### Euler characteristics and the difference bundle construction

I am reading on K theory in Lawson and Michelson (Spin Geometry). One has the "exact sequence spaces" $L(X,Y)$ and there is the theorem that there is a unique equivalence of functors $\chi$ between ...

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168 views

### non degenerate quadratic form on the group of correspondences on an algebraic curve?

Hi,
Given two (smooth, projective) curves $X$ and $Y$ over a field $k$, define a correspondence to be a line
bundle $L$ on $X\times Y$. A trivial correspondence is a correspondence of the form ...

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227 views

### 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 ...

**2**

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**1**answer

283 views

### Higher Euler characteristics (possible generalizations)

Let $X$ be projective and Gorenstein (over $\mathbb{C}$), of dimension $n$, then $\chi(\mathcal{O}_X)=(-1)^n\chi(w_X)$. Hence a "generalization": $\chi(w^{\otimes k}_X)$.
I'd like something of this ...

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141 views

### 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 ...

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149 views

### Is there some short formula for the “defect” of Hilbert function

Let $X\subset\Bbb P^n_{\Bbb C}$ be a connected, Cohen Macaulay sub-scheme. (Possibly singular, reducible or non-reduced.) For $k\gg0$ the numbers $h^0(\mathcal{O}_X(k))$ depend polynomially on $k$. ...

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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, ...

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### How to show the Euler Characteristic is equal to self-intersection number of zero-section [duplicate]

myThe definition of the Euler characteristic (given in Guillemin and Pollack's "Differential Topology") of a compact oriented manifold $X$ is the self-intersection number of the diagonal $\Delta$ in ...

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313 views

### The query concerning the Euler-Poincare formula’s generalizations

Euler's equation for polyhedral, Euler's polyhedral formula, V – E + F = 2, where V, E, and F, are the number of points, edges and faces, was discovered by Leonhard Euler in 1752. However, the basic ...