# Explicit computation of euler number of tangent bundle of sphere

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