27
votes
Accepted
A difficult integral for the Chern number
If Stokes' theorem counts as a standard technique, then here's an answer:
Introduce a "vector potential"
\begin{equation}
A_i = \frac{1-\hat n_z}{\hat n_x^2 + \hat n_y^2}(\hat n_x \partial_i ...
8
votes
Accepted
Odd integral Stiefel–Whitney classes in terms of even ones
$\DeclareMathOperator\Sq{Sq}$The odd classes $\beta(w_{2i})$ are part of your list of generators.
More interesting (and maybe what you wanted to ask?) is where the classes $\beta(w_{2i+1})$ are.
For ...
6
votes
Accepted
Why does Bott's obstruction theorem imply the vanishing of some cohomology classes of $B\Gamma_q$?
This is explained very nicely in [Law77]. Here is a sketch.
We will use the following crucial lemma.
Lemma (Haefliger). Let $M$ be a smooth manifold and let $\mathcal{H}$ be a $\Gamma_q$-structure on ...
5
votes
Induced fiber sequence and Eilenberg–MacLane space in Whitehead tower of $BO$
The takeaway is that the three classes $w_1$, $w_2$, and $\frac{1}{2}p_1$ are generators for their corresponding cohomology groups. This is the property one needs so that the homotopy fiber is the ...
4
votes
Accepted
R. Bott's lectures on characteristic classes
It turns out that what you are looking for is here
which was noted by Mostow and Perchik.
1
vote
Chern character of a super-connection (Heat kernels and Dirac operators)
After further reading I found that the issue is actually addressed on page 49:
Short answer:
The chern character is indeed a limit and for now I assume that the authors checked that the transgression ...
1
vote
Accepted
Necessary and sufficient conditions for pseudo Riemannian manifold to be time orientable
For the first question I believe the answer is yes. Almost certainly it's in the literature but I do not know this literature very well.
The idea is as you suggested. Maximal-rank timelike (or ...
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