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.
quantum invariants, integral basis, the even cobordism category, the Maslov index, modular categories
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