21
votes
Accepted
Source of a quote by Ferdinand Rudio
The quote is from a speech Rudio gave at the Town Hall in Zürich on the 6th December 1883; The German original is published in Felix Stähelin, Reden und Vorträge (1956, I have not found it online).
...
- 188k
13
votes
Accepted
Refined Euler characteristic
Look up the Poincare polynomial $p_X(t)$. It is still multiplicative, by the Kunneth formula. The Euler characteristic is $\chi_X=p_X(-1)$. We have $p_{S^1\times S^1}(t)=(t+1)^2$ but $p_{S^1\times [0,...
- 3,272
8
votes
Accepted
Compactification of open manifolds in the form of a manifold( with zero Euler characteristic)
First note that odd dimensions the question of Euler characteristic $0$ is automatic, $M$ will embed in the orientable double cover of $\tilde{M}$, which will have $\chi = 0$ by Poincare Duality.
In ...
- 6,995
7
votes
Multiplicativity of Euler characteristic for non-orientable fibrations
Since this question seems to be attracting some renewed interest, I may as well point out that a few years after I asked it, Kate Ponto and I proved a generalization of this formula by purely ...
- 66.8k
6
votes
Accepted
Euler Characteristic of $SL_m(\mathbb{C})/SO_m(\mathbb{C})$
Let $G$ be a connected complex reductive affine algebraic group, and let $H$ be an algebraic subgroup (assume reductive). Then the Euler characteristic of the homogeneous space $G/H$ can be computed ...
- 8,529
6
votes
Euler characteristic of the simplicial complex of sets of elements in a semilattice with non-zero meet
To get a complete answer, I think you will need to use the specific properties of your semilattices. But I think the key observation is to notice what the maximal faces of your complex $Z_c$ are:
A ...
- 1,846
6
votes
Accepted
Compact simply-connected homogeneous symplectic manifold
Let your manifold be $X=G/H$. First of all, since it is simply connected, we can write it as $K/U$ where $K$ and $U=K\cap H$ are compact in $G$ (Montgomery’s theorem, 1950). Next, since $K/U$ is ...
- 31.5k
5
votes
Accepted
Euler characteristic of local system depends only on rank?
As $X$ is proper, the Swan conductor of $\mathcal{F}$ vanishes. Hence the identity $\chi(X,\mathcal F)=\operatorname{rk}\mathcal F\cdot\chi(X,\underline{\mathbb F_\ell}_X)$ follows from Theorem $4.2.9$...
- 7,249
5
votes
Accepted
Euler characteristic of the simplicial complex of sets of elements in a semilattice with non-zero meet
This is a special case of the crosscut theorem. See e.g. Corollary 3.9.4 of Enumerative Combinatorics, vol. 1, second ed. Let $L'$ be $L$ with a top element $\hat{1}$ adjoined. In Corollary 3.9.4 take ...
- 50.8k
4
votes
When are bundles of odd and even differential forms isomorphic?
Strengthening your fourth bullet point, one can see all the Chern classes are equal using the complex splitting principle. So one cannot use Chern classes to distinguish them.
Indeed, I will prove the ...
- 149k
4
votes
Accepted
Euler characteristic of pseudomanifolds with boundary
Ok, to convert my comment to an answer. Let $S$ be a closed orientable triangulated surface of genus $\ge 1$. Let $M$ be the cone over $S$. Then $M$ has a natural orientable pseudomanifold structure. ...
- 12.3k
3
votes
Accepted
When are bundles of odd and even differential forms isomorphic?
I will explain that as long as $n>2$ the real vector bundles $\Omega^{even}$ and $\Omega^{odd}$ over $M$ are isomorphic.
If $n>2$ then $dim(\Omega^{even}) = dim(\Omega^{odd}) = 2^{n-1} > n$ ...
- 18.2k
3
votes
Accepted
Intuition for the Euler form in a finitary category
This answer perhaps says things that are all obvious to the OP.
$\textrm{Ext}^i(A,B)$ is a vector space over the ground field, so its cardinality is $q^d$ where $d$ is the dimension of the vector ...
- 6,327
2
votes
Accepted
Variation of Euler characteristic when the sheaf is not flat
Kollar in his article mentions that in general Euler characteristic is lower semi-continuous without the assumption of flatness. However, there is no proof of this statement in the article.
- 2,032
2
votes
Accepted
Is this line of thought (using linear algebra to get number theoretic results) already being pursued in the literature?
I don't know of such a reference, but I think I can give you a short proof for your intuition and conjecture, hope that helps:
One basis for $1,...,n$ is given by $e_1,\ldots, e_k$, where $k = \Pi(n)$...
- 809
2
votes
For which classes of topological spaces Euler characteristics is defined?
The Euler Characteristic can be defined for a larger class of spaces. For example, Euler Characteristic is defined for definable sets in an o-minimal system such as: semi-linear sets, semi-algebraic ...
- 909
1
vote
On the notion of multiplicity of a fixed point
Note that the fixed points of $f$ can be characterized as the intersection points of the graph of $f$, which is a submanifold of $M\times M$, with the diagonal of $M\times M$
$$\Delta_M=\big\{ (x,...
- 34.7k
1
vote
Multiplicativity of Euler characteristic for non-orientable fibrations
Let $B$ be a finite simplicial complex. Let us consider its dual cell-decomposition.
The preimage of each cell has Euler characteristic equal to that of $F.$
Now use the usual combinatorial formula ...
- 2,339
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