Questions tagged [euler-characteristics]

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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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9 votes
1 answer
392 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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  • 389
3 votes
1 answer
177 views

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 ...
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7 votes
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162 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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Is the secondary Euler characteristic a categorical trace?

Context: The ordinary Euler characteristic of a complex (satisfying appropriate finiteness conditions so that all cohomology groups are finite-dimensional over some field ''k'', say, and only finitely ...
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  • 1,704
8 votes
2 answers
215 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 $...
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3 votes
1 answer
323 views

Variation of Euler characteristic when the sheaf is not flat

Let $f:X \to Y$ be a flat, projective morphism with $Y$ integral and every fiber of $f$ normal and integral. Let $F$ be a torsion-free, coherent sheaf on $X$ (not necessarily flat over $Y$). Then, is ...
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452 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 $...
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12 votes
1 answer
509 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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11 votes
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618 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) \...
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A version of Hurwitz' theorem in terms of Euler characteristic

Page 203 of Farb and Margalit's Primer on Mapping Class Groups contains the result: Let $g ≥ 2$. The order of any finite subgroup of $MCG(S_g)$ is at most $84(g − 1)$. I've been told by my ...
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2 votes
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78 views

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 ...
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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 ...
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2 votes
1 answer
365 views

Euler Characteristic of $SL_m(\mathbb{C})/SO_m(\mathbb{C})$

As described in the title, what is the (topological) Euler characteristic of the homogeneous space $SL_m(\mathbb{C})/SO_m(\mathbb{C})$?
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Is this line of thought (using linear algebra to get number theoretic results) already being pursued in the literature?

Let $Log(n) = \sum_{i=1}^r \alpha_i \cdot e_i$, where $n = \prod_{i=1}^r p_i^{\alpha_i}$ and $p_i$ is the $i$-th prime, $\alpha_i \ge 0$, $e_i$ is the $i$-th standard basis vector. For example $6 = 2\...
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8 votes
1 answer
226 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 ...
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  • 369
0 votes
1 answer
215 views

On the notion of multiplicity of a fixed point [closed]

I am trying to understand the notion of multiplicity of a fixed point of a map $f: M \to M$, say, $M$ being a smooth closed manifold, and $f$ being a smooth diffeomorphism. There is a notion of ...
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1 vote
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370 views

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 ...
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  • 1,157
3 votes
1 answer
192 views

Intuition for the Euler form in a finitary category

Suppose that $\mathcal{C}$ is a finitary category, so for any two objects $A$ and $B$ we have that $|\mathrm{Ext}^i(A,B)| < \infty$ for $i\geq 0$, suppose $\mathcal{C}$ has finite global dimension, ...
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7 votes
1 answer
561 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 $\tilde{M}$ ...
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3 votes
0 answers
304 views

Integral of Gaussian curvature multiplied by mean curvature

Let $M$ be a 3-manifold with positive definite metric $g$, and let $S\subset M$ be an oriented 2-surface. For $x\in S$ let $K(x)$ be the Gaussian curvature and $H(x)$ be the mean curvature of $S$ at ...
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5 votes
1 answer
620 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, ...
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1 vote
0 answers
418 views

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, ...
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What is the significance of the $-1$-simplex?

The number of $k$-simplex elements in an $n$-simplex is counted by the binomial coefficient $\binom{n+1}{k+1}$. For example, the $3$-simplex is the tetrahedron, which has the following elements: $4$ ...
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3 votes
0 answers
166 views

Non-multiplicative Euler-Poincaré Characteristics

Are there known examples of a non-multiplicative Euler-Poincaré characteristic on varieties? Let $\mathbf{Var}/k$ be the category of varieties over a filed $k$, i.e. the category of reduced separated ...
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0 votes
1 answer
187 views

Triviality of certain vector bundles

Let $M$ be a smooth manifold and let $SM$ be the bundle of symmetric bi-linear forms on $TM.$ Riemannian metrics are a particular kind of sections in this bundle. Since any manifold admits a global ...
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6 votes
1 answer
444 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 ...
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12 votes
1 answer
755 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 $U$. ...
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11 votes
3 answers
825 views

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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5 votes
1 answer
637 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 ...
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2 votes
0 answers
200 views

The Euler characteristic of Hilbert series

The Hilbert series of a graded vector space $V=\bigoplus_{n\mathbb Z}V_n$ is the (ordinary) generating function of the dimensions of its homogeneous components, $h_V(t)=\sum_{n\in\mathbb Z}t^n\dim V_n$...
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5 votes
1 answer
834 views

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{...
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4 votes
1 answer
235 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 $C \rightarrow \mathbb P^1$ ...
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6 votes
1 answer
449 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)...
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6 votes
0 answers
253 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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2 votes
0 answers
1k 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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32 votes
3 answers
4k views

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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1 vote
0 answers
398 views

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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25 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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  • 8,478
10 votes
2 answers
656 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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12 votes
1 answer
806 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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9 votes
1 answer
522 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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  • 3,022
5 votes
2 answers
640 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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3 votes
1 answer
353 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 $L$...
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3 votes
1 answer
583 views

Higher Euler characteristics (possible generalizations)

Let $X$ be projective and Gorenstein (over $\mathbb{C}$), of dimension $n$, then $\chi(\mathcal{O}_X)=(-1)^n\chi(\omega_X)$. Hence a "generalization": $\chi(\omega^{\otimes k}_X)$. I'd like ...
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0 votes
1 answer
525 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 ...
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16 votes
3 answers
2k 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 ...
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7 votes
2 answers
621 views

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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3 votes
1 answer
187 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 $p_1^*...
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  • 637
14 votes
3 answers
2k 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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