All Questions
Tagged with cohomology vector-bundles
30
questions
2
votes
0
answers
107
views
Correct notion of "connected" for dga of bundle-valued forms
Consider a vector bundle $E$ over a manifold $M$ with flat connection, $\nabla$. From this data I can form the associative/unital differential graded algebra $\mathcal{A} = \left(\Omega^{\bullet}(M, ...
- 1,426
1
vote
0
answers
179
views
Vector bundles are first cohomology sets
I posted the following question on math.stackexachange and it was suggested that I should repost it here.
Suppose $X$ is a variety, then we have the result that $\operatorname{H}^1(X, \mathcal{O}^*_X)=...
- 821
3
votes
0
answers
198
views
Splitting vector bundles on $\mathbb{P}^n$
There are results by Kleiman and Sumihiro that claim you can split an algebraic vector bundles (in the sense that it admits filtration that quotients are given by line bundles) by applying ...
- 5,861
4
votes
0
answers
228
views
Relative Thom isomorphism
Let $\tilde{X}$ be a space with an action of the symmetric group $\mathfrak{S}_k$ and define $X:=\tilde{X}/\mathfrak{S}_k$ to be the quotient. On the other hand, $\mathfrak{S}_k$ acts on $(\mathbb{R}^...
- 1,623
2
votes
0
answers
138
views
Definition of odd topological K-theory using circles
I wanted to check whether the following characterization of odd complex topological $K$-theory is correct (reposted from Math.SE).
Let $X$ be a compact Hausdorff space. Then $K^{-1}(X)$ can be defined ...
- 1,901
2
votes
0
answers
96
views
lie algebra bundle and underlying vector bundle
Let $G$ be a connected reductive group over a field $k$. Let $E$ be a $G$-bundle, then we can form the adjoint bundle $ad(E)$ which is a Lie algebra bundle over $k$.
As a vector bundle it is trivial, ...
- 3,452
1
vote
0
answers
203
views
Find torsion classes using flat bundles
My question refers to a discussion from this older thread on Neron-Severi group of a Kähler manifold. In the comments below Ted Shifrin's answer there arose a discussion when the map $H^2(X,\mathbb{Z}...
- 5,780
3
votes
1
answer
865
views
Chern classes of complex vector bundle
I'm reading characteristic classes form the book Differential forms in Algebraic Topology by Bott and Tu. The Chern classes are defined as follows:
$E\xrightarrow{\rho} M$ is a vector bundle and $E_p$...
4
votes
0
answers
107
views
Generalized de Rham cohomology on product bundle giving specified cohomology
Given a compact, smooth manifold $M$ and a real vector bundle $E \to M$ (in general not flat). There already have been numerous questions about how to equip the space $\bigoplus_k \Gamma(\Lambda^k T^* ...
7
votes
1
answer
741
views
What is the geometrical meaning of higher Chern forms and classes?
Let $M$ be a complex manifold, $R^{\nabla}$ be the curvature operator for connections $\nabla$.
Consider a polynomial function $f:\operatorname M_n(\mathbb{C})\to\mathbb{C}$. For the gauge group $\...
- 321
5
votes
0
answers
254
views
Atiyah class and coboundary map
Let $L$ be a line bundle on a smooth algebraic variety $X$. Let $\sigma_i:U_i \times \mathbb{C} \to L_{|U_i} $
be its local trivializaations and $u_{ij}$ the transition functions satisfying $\sigma_j=...
- 3,737
7
votes
1
answer
538
views
First Chern class of a specific line bundle
Let $E$ be a spin$^c$ bundle and $spin^c(E)$ the corresponding $spin^c(n)$-principial bundle. Let $g_{U,V}: U \cap V \to spin^c(n)$ denote transition functions for this principial bundle and consider ...
- 9,240
7
votes
1
answer
551
views
Continuous maps $f:S^n \to \mathbb{C}P^m$ with $f(x)\perp f(-x) $
Question 1: What is a complete classification of all positive integers $m,n$ with the following property:
There is a continuous map $f:S^n \to \mathbb{C}P^m$ such that $f$ maps antipodal ...
- 452
4
votes
0
answers
188
views
Obstruction to the existence of lifting of the classifying map
Let $E$ be an $n$-plane bundle over CW complex $X$. Then $E$ is a pullback of tautological bundle $\gamma_n$ over $BO(n)$ i.e. $E=f^*(\gamma_n)$. This $f$ is called classyfing map. One can show that ...
- 9,240
5
votes
2
answers
334
views
Two set of axioms for Stiefel-Whitney classes
Let $E \to X$ be a vector bundle. We can associate to $E$ several invariants: among them are the Stiefel-Whitney classes $w_i(E) \in H^i(X;\mathbb{Z}_2)$. These classes may be defined using the axioms:...
- 9,240