In a sense, calculus is all about the study of critical points of functions on flat space $\mathbb{R}^N$ (e.g. here). Let's try a different venue, the unit tangent bundle of the sphere. $$ T^1(S^2) = \big\{ (x,\vec{v}): x \in S^2,\, \vec{v} \in T_x(S^2),\, \big|\big|\,\vec{v}\,\big|\big|=1 \big\} $$

And there are other example of circle bundles over the sphere, such as tensor products of this bundle with itself. I was able to obtain on Math.SE a rather general result: cohomology of circle bundles can be found via the Gysin exact sequence. Can obtain this result via Morse theory?

- $H^0\big( T^1(S^2) \big) \simeq \mathbb{Z} $
- $H^1\big( T^1(S^2) \big) \simeq 0 $
- $H^2\big( T^1(S^2) \big) \simeq \mathbb{Z}/2\mathbb{Z} $
- $H^3\big( T^1(S^2) \big) \simeq \mathbb{Z} $

I post this question here because I know explicit Morse functions always exist but can be difficult to find. And I'm trying to work out the critical points and visualize the Morse flow.

And perhaps I should clarify what I mean by "explicit". I believe the unit tangent bundle could be given the structure of a variety. Looking at our construction, we could try to embed

$$ T^1(S^2) \subseteq \mathbb{R}^3 \times \mathbb{R}^3$$

I don't think his is quite the "universal bundle" construction, but it's something. And we could write down the constraints:

- $x_1^2 + x_2^2 + x_3^2 = 1$
- $v_1^2 + v_2^2 + v_3^2 = 1$
- $\vec{v} \in T_x(S^2) \subseteq \mathbb{R}^3$ which could be a hyperplane in 3-space.
- $\langle x, v \rangle = x_1 v_1 + x_2 v_2 + x_3 v_3 = 0$ this shows that $\vec{x} \perp \vec{v} $ and that $\vec{v}$ is tangent to $S^2$.

And if we pin down all the relations we obtain the structure of an algebraic variety. Therefore, could it be possible to write down **polynomial** morse functions of this kind of space? And work out the critical points?

- How many polynomial Morse functions on the sphere?
- cohomology module of unit tangent vector bundles over spheres
- https://en.wikipedia.org/wiki/Gysin_homomorphism

Even a derivation of the Gysian homomorphism via Morse theory could be interesting. Certainly I've never seen it.

We could try to build Morse functions of out of polynomials in $x$ and $v$. Does the set Morse functions of given degree form a vector space?