Questions tagged [euler-characteristics]

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32 votes
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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 ...
Jochen Wengenroth's user avatar
32 votes
4 answers
3k views

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 ...
Ilya Nikokoshev's user avatar
27 votes
2 answers
3k 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 ...
Dick Palais's user avatar
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25 votes
0 answers
1k 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 ...
Vivek Shende's user avatar
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23 votes
6 answers
2k 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 ...
Theo Johnson-Freyd's user avatar
20 votes
2 answers
1k views

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 ...
Qiaochu Yuan's user avatar
18 votes
4 answers
2k 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 ...
John Baez's user avatar
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17 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 ...
Raghav Kulkarni's user avatar
14 votes
4 answers
4k 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 ...
Ariyan Javanpeykar's user avatar
14 votes
3 answers
3k 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 ...
Mike Shulman's user avatar
13 votes
2 answers
562 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$ ...
Ceka's user avatar
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12 votes
1 answer
518 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 ...
Francois Ziegler's user avatar
12 votes
1 answer
796 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 $U$. ...
Dominik's user avatar
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12 votes
1 answer
927 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 ...
Vivek Shende's user avatar
  • 8,663
11 votes
3 answers
945 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 ...
Brenin's user avatar
  • 1,514
11 votes
2 answers
1k 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 ...
Philipp Lampe's user avatar
11 votes
1 answer
333 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. ...
Reid Barton's user avatar
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11 votes
0 answers
627 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) \...
user avatar
10 votes
2 answers
2k 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 $\chi$...
Dmitri Panov's user avatar
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10 votes
2 answers
686 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 ...
Theo Johnson-Freyd's user avatar
10 votes
2 answers
1k views

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 ...
Dylan Wilson's user avatar
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9 votes
1 answer
2k 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 (...
Jesko Hüttenhain's user avatar
9 votes
1 answer
553 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 ...
macbeth's user avatar
  • 3,182
8 votes
2 answers
250 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 $...
Christian Gorski's user avatar
8 votes
1 answer
234 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 ...
Valentino's user avatar
  • 369
8 votes
1 answer
472 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 ...
Dominik's user avatar
  • 3,007
7 votes
1 answer
487 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 $...
James Propp's user avatar
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7 votes
3 answers
1k 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 ...
Chen's user avatar
  • 381
7 votes
2 answers
661 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 ...
Alexey Muranov's user avatar
7 votes
1 answer
610 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 $\tilde{M}$ ...
Ali Taghavi's user avatar
7 votes
0 answers
176 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 $X(\mathbb C)$ ...
Asvin's user avatar
  • 7,648
6 votes
1 answer
501 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)...
Will Sawin's user avatar
  • 135k
6 votes
0 answers
281 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 ...
Raghav Kulkarni's user avatar
5 votes
1 answer
668 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 ...
Theo Johnson-Freyd's user avatar
5 votes
2 answers
713 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 ...
Jesko Hüttenhain's user avatar
5 votes
1 answer
927 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{...
user avatar
5 votes
1 answer
741 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, ...
John Pardon's user avatar
  • 18.3k
5 votes
2 answers
1k 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 ...
Sam Lewallen's user avatar
  • 1,109
5 votes
0 answers
195 views

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 "(...
B.Hueber's user avatar
  • 987
4 votes
3 answers
1k 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 ...
Sam Lewallen's user avatar
  • 1,109
4 votes
1 answer
1k 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 ...
Ariyan Javanpeykar's user avatar
4 votes
1 answer
274 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 $C \rightarrow \mathbb P^1$ ...
Davide Cesare Veniani's user avatar
4 votes
1 answer
885 views

"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, ...
Theo Johnson-Freyd's user avatar
4 votes
0 answers
162 views

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 ...
Simona Vesela's user avatar
4 votes
0 answers
190 views

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 ...
Neil Strickland's user avatar
4 votes
0 answers
95 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 ...
Chris H's user avatar
  • 1,854
4 votes
0 answers
205 views

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 ...
Asvin's user avatar
  • 7,648
4 votes
0 answers
312 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 ...
Allen Knutson's user avatar
3 votes
1 answer
660 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 ...
Dmitry Kerner's user avatar
3 votes
1 answer
412 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 ...
G. Blaickner's user avatar
  • 1,137