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 ...
Michał Jan's user avatar
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 ...
Jan Nienhaus's user avatar
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 ...
Ken's user avatar
  • 1,749
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 ...
Michael Albanese's user avatar
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.
N.Li's user avatar
  • 92
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 ...
Filippo's user avatar
  • 329
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 ...
Ryan Budney's user avatar
  • 42.8k

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