In chapter 1, which represents joint work with Gilmer, we define an index two subcategory of a 3-dimensional cobordism category. The objects of the category are surfaces equipped with Lagrangian subspaces of their real first homology. This generalizes the result of [9] where surfaces are equipped with Lagrangian subspaces of their rational first homology. To define such subcategory, we give a formula for the parity of the Maslov index of a triple of Lagrangian subspaces of a skew symmetric bilinear form over R.
In chapter 2, we find two bases for the lattices of the SU(2)-TQFT-theory modules of the torus over given rings of integers. We find bases analogous to the bases defined in [13] for the lattices of the SO(3)-TQFT-theory modules of the torus. Moreover, we discuss the quantization functors (V_{p}, Z_{p}) for p = 1, and p = 2. Then we give concrete bases for the lattices of the modules in the 2-theory. We use the above results to discuss the ideal invariant defined in [7]. The ideal can be computed for all the 3-manifolds using the 2-theory, and for all 3-manifolds with torus boundary using the SU(2)-TQFT-theory. In fact, we show that this ideal using the SU(2)-TQFT-theory is contained in the product of the ideals using the 2-theory and the SO(3)-TQFT-theory under a certain change of coefficients, and it is equal in the case of torus boundary.
In chapter 3, we give a congruence which relates the quantum invariant of a prime-periodic 3-manifold to the quantum invariant of its orbit space. We do this for quantum invariant that is associated to any modular category over an integrally closed ground ring.
