Quantum invariants, skein modules, and periodicity of 3-manifolds

We study various quantum invariants and skein modules of 3-manifolds and their relationship to each other and to classical invariants of 3-manifolds. We study, in particular, how quantum invariants and classical invariants (homology) reflect periodicity of 3-manifolds. The first chapter is devoted to Turaev-Viro invariants. We show that each Turaev-Viro invariant is a sum of three invariants, study properties of the summand invariants and establish certain relationships. We solve a conjecture due to Kauffman and Lins and provide a simple and effective criterion answering whether two given lens spaces are distinguished by Turaev-Viro invariants or not. Chapter 2 is probably the most interesting chapter of the theses. We develop a special surgery presentation for p-periodic 3-manifolds and prove the following result: if a closed orientable 3-manifold M admits an action of a cyclic group Z p where p is an odd prime integer and the fixed point set of the action is S1 then H 1 (M; Zp) ≠ Zp. We also provide a similar criterion for the 2-periodic rational homology 3-spheres. Chapter 3 studies how quantum invariants reflect periodicity of 3-manifolds. We consider Dijkgraaf-Witten invariants and the simplest known quantum invariants discovered by H. Murakami, T. Ohtsuki, and M. Okada. In the last chapter, Chapter 4, we study some simple skein modules ( S and S2 ) and their relationship with quantum invariants. We show how to construct Murakami-Ohtsuki-Okada invariants from these skein modules using Lickorish's method. In Section 3 of Chapter 4 we will show that if q is a root of unity then no other invariants can be obtained using Lickorish method on S and S2 . We also study Deloup invariants and their relationship with the tensor product of two copies of S2 (M; R, q). In Appendix A we show how to define Dijkgraaf-Witten invariants using singular triangulations and simple 2-skeletons of 3-manifolds. In Appendix B we present tables with values of Turaev-Viro invariants for 239 closed oriented 3-manifolds of small complexity. The values are presented as cyclotomic integers. In Appendix C we show how to determine linking pairing of a rational homology 3-sphere from its rational surgery presentation.