Let $M = H_\alpha \cup H_\beta$ be a closed connected Heegaard split 3-manifold with $\Sigma = H_\alpha \cap H_\beta$. Let $\tilde{M}$ be the universal cover of $M$ and let $\tilde{\Sigma}$ be the lift of $\Sigma$ to $\tilde{M}$. Similarly let $\tilde{H_\alpha}$ and $\tilde{H_\beta}$ be the lifts of $H_\alpha$ and $H_\beta$.
What can I say about the topology of $\tilde{\Sigma}$? Is it necessarily connected? What is the subgroup of $\pi_1(\Sigma)$ that $\tilde{\Sigma}$ corresponds to? I imagine that it can be written in terms of the kernels of the inclusions $\pi_1(\Sigma) \to \pi_1(H_\alpha)$ and $\pi_1(\Sigma) \to \pi_1(H_\beta)$?
What about $\tilde{H_\alpha}$ and $\tilde{H_\beta}$? In the case where the cover is finite-sheeted is $\tilde{M} = \tilde{H_\alpha} \cup \tilde{H_\beta}$ a Heegaard splitting?
I would really love a reference that addresses these sorts of ideas. I am sorry for such an open ended question - I am not really initiated in the 3-manifold world and I do not know where to look for these things.