Genealogy of Nonperturbative Quantum-Invariants of 3-Manifolds: The Surgical Family

We study the relations between the invariants $\tau_{RT}$, $\tau_{HKR}$, and $\tau_L$ of Reshetikhin-Turaev, Hennings-Kauffman-Radford, and Lyubashenko, respectively. In particular, we discuss explicitly how $\tau_L$ specializes to $\tau_{RT}$ for semisimple categories and to $\tau_{HKR}$ for Tannakian categories. We give arguments for that $\tau_L$ is the most general invariant that stems from an extended TQFT. We introduce a canonical, central element, {\sf Q}, for a quasi-triangular Hopf algebra, $\A$, that allows us to apply the Hennings algorithm directly, in order to compute $\tau_{RT}$, which is originally obtained from the semisimple trace-subquotient of $\A-mod$. Moreover, we generalize Hennings' rules to the context of cobordisms, in order to obtain a TQFT for connected surfaces compatible with $\tau_{HKR}\,$. As an application we show that, for lens spaces and $\A=U_q(sl_2)\,$, the ratio of $\tau_{HKR}$ and $\tau_{RT}$ is the order of the first homology group. In the course of this paper we also outline the topology and the algebra that enter invariance proofs, which contain no reference to 2-handle slides, but to other moves that are local. Finally, we give a list of open questions regarding cellular invariants, as defined by Turaev-Viro, Kuperberg, and others, their relations among each other, and their relations to the surgical invariants from above.