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This is related to Bott-Tu Sec 11, Sphere Bundles. I would like to compute the euler characteristic of sphere bundle concretely. I have figured out the following so far.

Let $S^k$ be the $k-$sphere. Consider vector field $v$ starting from south pole to north pole by flowing. Clearly this gives a section to $T(S^k)$. Now normalize this section by considering any riemannian metric $||\cdot||$ by $\frac{v}{||v||}$ and this gives a section on sphere bundle of $S^k$ with fiber $S^{k-1}$ away from 2 points(i.e. south and north pole).

Since Euler number can be computed by sum of local degrees at south pole and north pole, I can compute the local degree at local trivializations of sphere bundle at south pole $a$ and north pole $b$. WLOG, one can assume local trivializations look like $D(x)\times S^{k-1}$ where $D(x)$ is some open disk on sphere $S^k$ centered at $x$. Then consider the following maps. $\partial D(x)\xrightarrow{s}D(x)\times S^{k-1}\xrightarrow{\pi_2} S^{k-1}$ and denote total map as $f_x$. Pick volume form $\omega$ on $S^{k-1}$ where $\omega$ can be standard volume form.

Since the degree is computed locally, one can consider $\int_{\partial D(x)}f_x^\star(\omega)=\int_{\partial D(x)}s^\star\psi$ where $\psi$ is the global angular form defined over $E$ and $x$ can be either south or north pole. This follows from $f_x^\star$ defines form restricts cohomology generator on each fiber and thus same cohomology class as global angular form. Hence they differ by a total differential and then one can apply stoke theorem to see the integral agrees.(Here I have already assumed the support of $\psi$ is away from $D(x)$ and one can shrink the size of disk if necessary.)

If I denote $s(x)=(x,v(x))$ with $v(x)\in S^{k-1}$, then $\pi_2\circ s(x)=v(x)$. Hence $\int_{\partial D(x)}f_x^\star(\omega)=\int_{\partial D(x)} v^\star(\omega)$. I can take $\omega$ to be standard volume form of $k-1$ sphere.

$\textbf{Q:}$ Now I do not see an obvious way to compute this integral though I can write down the formula. How do I proceed further?

Ref. Bott-Tu Differential Forms in Algebraic Topology, Sec 11, Thm 11.16, Exercise 11.21

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