All Questions
31
questions
2
votes
0
answers
138
views
Compute the Euler class of tautological $C$-bundle over $CP^1$
$\DeclareMathOperator\SO{SO}$This might be an old question. But since I have not found an explicit answer to this question, I put the question here.
The background is that we need to use a similar ...
- 21
11
votes
3
answers
1k
views
Computation on characteristic classes
I am organizing a reading seminar on characteristic classes. The audience in the seminar is interested in symplectic and contact manifolds. I work in categorification and would like to compute some ...
- 331
7
votes
1
answer
399
views
Meaning of the first Chern class of the unit tangent bundle of a surface
(This is a fairly basic question, not really research level, except that I am a research mathematician working on other things who is trying to understand more topology for use in my own work.)
Let $\...
- 121
5
votes
0
answers
119
views
How to see $\delta_2(\hat{\chi}(V))=\chi(V)$ in differential cohomology?
I'm reading the paper "Differential Characters and Geometric Invariants" by Cheeger and Simons. In Page 62 the authors defined the differential Euler character $\hat{\chi}(V)\in \hat{H}^{2n-...
- 8,767
13
votes
2
answers
577
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$ ...
- 501
0
votes
0
answers
257
views
Define a characteristic class on a simplicial complex (non-manifold)
Given a simplicial complex with only triangulation and only branching structure, is it enough to define Stiefel–Whitney class?
(Please provide Yes or No answers, and reasonings.)
Given a fixed ...
- 10.4k
4
votes
0
answers
140
views
Characteristic classes of quotient manifold
Let $M$ be a compact oriented smooth manifold with boundary and let $G$ be a compact Lie group acting smoothly, orientation-preservingly and freely on $M$.
(Under what conditions) is there a ...
- 641
9
votes
0
answers
336
views
Is $\beta^{*}(w_{2k-2}) = 0$ for an open orientable $2k$-manifold?
This question is motivated by the vector field question I asked recently. Panagiotis Konstantis answered this question for odd manifolds and I am trying to figure out the even case.
Let $M$ be a ...
- 1,646
2
votes
1
answer
549
views
Relation between compact vertical cohomology and local cohomology groups
I'm reading the books by Bott & Tu and Milnor & Stasheef simultaneously. The following is my doubt:
The Thom isomorphism in Bott & Tu is obtained as $H_{cv}^{*+n}(E)\rightarrow H^*(M)$, ...
5
votes
1
answer
369
views
Conversion formula between "generalized" Stiefel-Whitney class of real vector bundles: O(n) and SO(n)
$O(n)$ is an extension of $\mathbb{Z}_2$ by $SO(n)$,
$$1\to SO(n) \to O(n)\to \mathbb{Z}_2 \to 1.$$
Below we denote the Stiefel-Whitney class of real vector bundle $V_G$ of the group $G$ as:
$$
w_j(...
- 10.4k
7
votes
1
answer
390
views
Characteristic classes of the bundle of trace free, skew adjoint endomorphisms
In "Floer Homology groups in Yang-Mills theory", Donaldson says that if we take an $U(2)$-vector bundle $E$ and we construct the bundle $\mathfrak{g}_E$ of trace-free, skew adjoint automorphisms of $...
- 2,533
3
votes
0
answers
169
views
Classification of fibrations for classifying spaces $B^2\mathbb{Z}_2$ and $BSO(2)$ or $BO(2)$
Thanks to a suggestion by @Igor Belegradek, I am interested also in a simpler problem of this earlier question 301523, by knowing what can we say about the classification of fibrations for classifying ...
- 2,147
3
votes
1
answer
562
views
How does one introduce characteristic classes [closed]
How does one introduce, or how were you introduced to characteristic classes?
You can assume that the student is comfortable with principal bundles and connections on principal bundles.
I am not ...
12
votes
2
answers
993
views
A Compact Manifold with odd Euler characteristic whose tangent bundle admits a field of lines
I understand that the top Stiefel Whitney class is an obstruction for the tangent bundle of a manifold to have a trivial line sub-bundle. I am looking for a counterexample when removing the word "...
- 953
12
votes
2
answers
589
views
Steenrod powers of Pontryagin classes
It is well known that the Stiefel–Whitney classes $w_i$ of a smooth manifold are generated, over the Steenrod algebra, by those of the form $w_{2^{i}}$. I wonder if it the same statement is known/true ...
- 1,420