Homology of Unit Tangent Bundle of a Riemann Surface

Question

Let $X$ be a Riemann surface and let $UT(X)$ be its unit tangent bundle.

Could someone help me understand the relation between their homology groups $H_1(X)$ and $H_1(UT(X))$, please?

Thank you!

Answer 1

You can exploit the fact that the 1-domensional homology group of a space is the abelianisation of the fundamental group of the space, while the fundamental groups are the 1-homotopy groups, so that they fit in a long exact sequence $\pi_2(S^1)\to \pi_2(UTX)\to \pi_2(X)\to \pi_1(S^1)\to \pi_1(UTX)\to \pi_1(X)\to$, etc. This should be enough for most calculations. Note that $\pi_2(S^1)=0$, and also $\pi_2(X)=0$ unless $X$ is the 2-sphere or the real projective plane.