in quantum topology

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Abstract
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)−TQFTtheory 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 primeperiodic 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.
v Chapter 1 The Parity of the Maslov Index and the Even Cobordism Category1

Introduction
In [9], Gilmer considered a cobordism category C.This category can be described roughly as follows.The objects of C are closed surfaces equipped with Lagrangian subspaces of their rational first homology.A morphism of C N : Σ → Σ is a cobordism between Σ and Σ .Also, he defined a subcategory C + of C of index two.It would be more consistent with other work [22] [23] to consider a similarly defined cobordism category C where the extra data of a Lagrangian subspace is a subspace of the real first homology.The main goal of this chapter is define an analogous index two subcategory C + of C. We call C + the even cobordism category.If one restricts to this 'index two' cobordism subcategory, one may obtain functors, related to the TQFT functors defined by Turaev with initial data a modular category, but without taking a quadratic extension of the ground ring of the modular category as is sometimes needed in [22, p.76].
It is not possible to simply modify the proof given in [9] for the existence of C + to obtain a proof for the existence of C + .This is because not every real Lagrangian subspace can be realized as the kernel of the map induced on first homology by the inclusion of a surface to a 3-manifold which has the surface as its boundary.Only the subspaces which are completions of subspaces of the rational homology can be so realized.So another approach has to be used.We actually reduce the problem to the one already solved for C but this requires some new algebraic results.These algebraic results may be of independent interest.We prove the algebraic results in §1.2.This section is written without any appeal to topology.We derive the following congruence for the Maslov index, denoted µ: Theorem 1.1.1Let V be a symplectic vector space and λ 1 , λ 2 , and λ 3 be any three Lagrangian subspaces, then we have If λ 1 ∩ λ 2 = λ 2 ∩ λ 3 = λ 1 ∩ λ 3 = 0, this result follows from [17, 1.5.7]which gives a formula for the Maslov index in terms of a special form these Lagrangian subspaces must take in this case.We give a very different proof.Theorem 1.1.1 will be the key to proving that the morphisms of C + are closed under composition.In §1. 3, we describe the weighted cobordism categories C and C in greater detail.In §1.4,C + is defined.

Lagrangian Subspaces and the Maslov Index
Let V be a symplectic vector space, i.e.V is finite dimensional over R and endowed with a skew symmetric bilinear form ψ. This is the terminology used in [22].Note that we do not require that the form is nondegenerate.If A is a subspace of V , its annihilator, Ann(A), is the set of elements which pair under the form with all of A to give zero.If A and A are two subspaces, then [22,IV.A subspace A ⊂ V is said to be a Lagrangian subspace if A = Ann(A).The proof of Theorem (1.1.1)is given in this section after we give all the results needed in the proof.
To show that this new form is well-defined, let a 1 , a 2 ∈ (λ As is wellknown, a non-degenerate symplectic vector space must be even dimensional.Hence We have the following well-known proposition [22, IV.3.5]Proposition 1.2.1 Let λ 1 , λ 2 and λ 3 be three Lagrangian subspaces of V .Define a bilinear form , on (λ where a, b ∈ (λ 1 + λ 2 ) ∩ λ 3 and a = a 1 + a 2 ., is a well-defined symmetric bilinear form.
Proof.To show is , is well-defined, note that the decomposition a = a 1 + a 2 , where for i = 1, 2, 3 and ψ is skew symmetric, we have Hence the form is symmetric.
In general, , is degenerate.In fact, it is known that its annihilator contains It is shown in [14] that this is true in general.
Proof.Let W denote the annihilator of this form.It is clear that λ 1 ∩ λ 3 ⊂ W , also ∩λ 3 ) in V .Using equations (1.1) and (1.2), we have that Proposition 1.2.2For any pair of Lagrangian subspaces λ 1 , and λ 2 we have and Proof.The first formula follows by reducing it to the nonsingular case and We obtain the second congruence from and the first formula.
Proof.Since the annihilator of the form is (λ The result follows as the signature and the rank of a nondegenerate form agree modulo two. Proof of Theorem 1.1.1.By equation (1.5), we have and also have Hence by Theorem (1.2.1), the left hand sides of these two congruences are congruent.So their right hand sides must be congruent as well: The last equation is equivalent to The left hand side of this last equation is congruent to the Maslov index by Corollary 1.2.1, and hence the first formula follows.The second formula follows by equation (1.4).
The following results from [22] give some properties of the Maslov index and will be used in the next section.
Lemma 1.2.1 For any Lagrangian subspaces λ 1 , λ 2 , λ 3 of V , we have Definition 1.2.2Let V, V be symplectic vector spaces.The symplectic vector space −V is just V as a vector space with the opposite (minus) form.Also, the symplectic vector space −V ⊕ V is the direct sum of V and V as a vector space with the sum of the two forms of −V and V .

The Weighted Cobordism Categories
All 3-manifolds and surfaces in this chapter are assumed to be oriented and compact.
We define a weighted cobordism category C whose objects are surfaces Σ without boundary equipped with a Lagrangian subspace λ ⊂ H 1 (Σ, R).We will denote objects by pairs (Σ, λ).A cobordism from (Σ, λ) to (Σ , λ ) is a 3-manifold together with an orientation preserving homeomorphism (called its boundary identification) from its boundary to −Σ Σ .Here, and elsewhere, −Σ denotes Σ with the opposite orientation.
Two cobordisms are equivalent if there is an orientation preserving homeomorphism between the underlying 3-manifolds that commutes with the boundary identifications.
A morphism M : (Σ, λ) → (Σ , λ ) is an equivalence class of cobordisms from (Σ, λ) to (Σ , λ ) together with an integer weight.We denote morphisms by a single letter.We let w(M ) denote the weight of M .We let M denote the underlying 3-manifold of a representative cobordism.This is well defined up to homeomorphism respecting the boundary identifications.We call (Σ, λ) the source of M and (Σ , λ ) the target of M .
We let j M denote the inclusion Σ into M, and j M denote the inclusion Σ into M.
Here and sometimes below we ignore the boundary identifications for simplicity and we write as if Σ Σ were the boundary of M .
The boundary of this new 3-manifold is equipped with a boundary identification in the obvious way.The weight of the composition is given by the formula 1 .
The identity id (Σ,λ) : (Σ, λ) → (Σ, λ) is given by Σ × I with the weight zero and the standard boundary identification.This is called a cylinder.Any morphism C : (Σ, λ) → (Σ, λ ) with Σ × I as the underlying 3-manifold, and with the standard boundary identification will be called a skew-cylinder over Σ.
Proof.This follows immediately from the definitions.One needs that the Maslov index vanishes when two of the three Lagrangian subspaces coincides which follows by [22, p183], or Theorem (1.2.2).
If we make the same definitions but using Lagrangian subspaces in H 1 (Σ, Q), we obtain the cobordism category C studied in [9].The proof of the following result is closely related to the proof of [22, Lem.9.1.1].
Lemma 1.3.2C and C are categories.
Proof.To prove that C or C is a category, we need to satisfy the two category axioms.
For the first one, we claim that the cylinder C over (Σ, λ) with weight zero is the identity morphism for the object (Σ, λ).To prove the claim, let M be any morphism from (Σ, λ) to (Σ , λ ).Then M • C is homeomorphic to M , and The last equality follows as the form is zero by Theorem (1.2.2).Hence we obtain that the first axiom holds.We claim that the composition is associative which is the second axiom.To prove this claim, we let (M i , w i ) : to prove that they have the same weight, i.e.
We apply formula (1.6) to obtain Similarly; So if we compare the above two equations, then we need to prove that This follows from Lemma (1.2.3) and Lemma (1.2.1) using the facts [22, pp. 182] and in the case ), and Hence we obtain that the second axiom holds.Therefore we conclude that C and C are cobordism categories. As ) which arises in this way is called rational.In this way, we obtain a functor C → C.

The Even Cobordism Category
We repeat a definition from [9] except now we apply it to morphisms of C instead of where (M ) is one if exactly one of Σ and Σ is nonempty and otherwise (M ) is zero.
If a cobordism is not even, it is called odd.
We note that the inverse of an even skew-cylinder is even.
The first author showed that the composite of two even morphisms of C is again even [9, Theorem 7.2].The subcategory C + was defined to be the category with the same objects as C but with only even morphisms.In the rest of this section, we generalize this result to morphisms in C. Given this result, we define the subcategory C + to be the category with the same objects as C but with only the even morphisms.We would also get a subcategory if we left the (N ) term out of Definition 1.4.1.However the definition that we give is more natural from some points of view [9].
Proof.Apply the definition above.Proof.We first show that M • C is even.We need to show By assumption, we have that: and, So after we substitute (1.8), (1.9) and (1.10) into (1.7),we conclude that we need only prove: For any subspace δ of H 1 (Σ, R), we have that as and kernel of j M * is a subset of M * (λ ).Thus we have that dim + n where n is the dimension of kernel of j M * .Thus both sides of (1.11) are congruent to n.Hence, we obtain (1.7).
The proof that C •M is even follows formally from the first part, if we consider how the parity of a cobordism changes when we reverse the orientation of the underlying 3-manifold and reverse the roles of source and target.
Proof.It follows by lemma (1.3.1) that we can factor M as Hence M is even by two applications of Lemma (1.4.1).
Theorem 1.4.1 The composition of two even morphisms of C is again even.
Proof.Let M 1 , M 2 be two even morphisms and adopt the notations associated to M 1 and M 2 in §1.3.We need to show that M 2 • M 1 is an even cobordism.It suffices to even for some even skew-cylinders over C and C over Σ and Σ with rational Lagrangian subspaces for Σ and Σ .On the other hand we can write where C is an even skew-cylinder over Σ whose the target has a rational Lagrangian.We have that where Chapter 2 Integral Bases for Certain TQFT Modules of the Torus

Introduction
We let p denote an odd prime or twice an odd prime unless mentioned otherwise.Also, we let Σ denote a surface of genus g.Gilmer defined an integral TQFT-functor S p in [9] based on the integrality results of the SO(3)-and SU (2)-invariants in [19,18].This is a functor that associates to a closed surface Σ, a module S p (Σ) over a certain cyclotomic ring of integers O p .Moreover, Gilmer showed that these modules are free in the case of p is an odd prime.Gilmer and Masbaum constructed basis for S p (Σ) and gave an independent proof of freeness in this case.In addition, Gilmer showed that these modules are projective where p is twice an odd prime.In this chapter, we prove that the modules S p (S 1 × S 1 ) are free by constructing two explicit bases in the case p is twice an odd prime.In the 2-theory, we prove also that the modules S 2 (Σ) are free by constructing an explicit basis for any surface.
Frohman and Kania-Bartoszynska in [7] defined an ideal invariant of 3-manifolds with boundary using the SU (2)-TQFT-theory that is hard to compute.In fact, they make use of another ideal that they defined to give an estimate for this ideal.However, Gilmer and Masbaum in [12] computed an analogous ideal invariant using the SO(3)−TQFT-theory for 3-manifolds that are obtained by doing surgery along a knot in the complement of another knot.The computations depend entirely on the fact that bases are constructed for the integral lattices of the SO(3)-TQFT-theory modules [13,12] of the torus.Also, Gilmer and Masbaum gave a finite set of generators for this ideal in general.Based on our results in this chapter, we compute this ideal for the above 3-manifolds with torus boundary using the SU (2)-TQFT-theory.Also, we introduce a formula to give an estimate for the ideal using the SU (2)-TQFT-theory in terms of the ideals using the 2-and SO(3)-TQFT-theories.In fact, the same formula can be used to compute this ideal using the SU (2)-TQFT-theory for the all the above 3-manifolds with torus boundary.
In §2.1, we describe the SO(3)-and SU (2)-TQFT-functors using the approach of [3] over a variant ring depending on p.We review the integral TQFT-functors in §2.2 that Gilmer defined in [9].The first bases for the lattices of the SU (2)-TQFT-modules are given in §2.3.We review the Frohman Kania-Bartoszynska ideal in §2. 4, and then we draw some conclusions based on the results of the previous section regarding this ideal.
The quantization functors for p = 1 and p = 2 are discussed in §2.5, again following [3].Also in this section, we give basis for S 2 (Σ), and then draw some conclusions regarding the Frohman and Kania-Bartoszynska ideal for this theory.We reformulate some results given in [3] in §2.6 to serve our need.Finally, we give another bases for the lattices of the SU (2)-TQFT-modules in §2.7.The advantage of this one over the first basis is that it allows us to prove Theorem (2.8.3).

The SO(3)-and SU (2)-TQFTs
We consider the (2+1)-dimensional TQFT constructed as the main example of [3, P. 456] with some modifications.In particular, we use the cobordism category C discussed in [9,14] where the 3-manifolds have banded links but surfaces do not have colored points.Hence the objects are oriented surfaces with extra structure (Lagrangian subspaces of their first real homology).The cobordisms are equivalence classes of compact oriented 3-manifolds with extra structure (an integer weight) with banded links sitting inside of them.Two cobordisms with the same weight are said to be equivalent if there is an orientation preserving diffeomorphism that fixes the boundary. Let Finally, it is known that V p is generated over k p by all connected vacuum states.
The modules V p (Σ) are free modules over k p , and carry a nonsingular Hermitian sesquilinear form given by (2.1) Here -M is the cobordism M with the orientation reversed and multiplying the integer weight by -1, and leaving the Lagrangian subspace on the boundary the same.
).One has that V p (S 1 × S 1 ) ∼ = k p [z]/I where the ideal I is generated by e dp − e dp−1 in the case of p is an odd prime and by e dp in the case of p is twice an odd prime (See [1] for more details).Thus indeed, V p (S 1 × S 1 ) has a basis {e 0 , . . ., e d p −1 } of rank d p .

The Integral Cobordism Functor
Let C be the subcategory of C consisting of the nonempty connected surfaces and connected cobordisms between them.Let O p be the ring of integers of the ring k p defined before.The ring of integers is given by Thus the ring of integers of k p is a Dedekind domain.
Definition 2.3.1 For the surface Σ, we define S p (Σ) to be the O p -submodule of V p (Σ) generated by all connected vacuum states.
Hence we obtain a functor from C to the category of O p -modules.These modules are projective as they are finitely generated torsion-free over Dedekind domains [9, Thm.2.5].Also, these modules carry an O p -Hermitian sesquilinear form ( , given by The value of this form always lies in O p by the integrality results for closed 3manifolds in [19,18].(discussed in §5) for all surfaces and for S 1 × S 1 in the case of p is twice an odd prime.
A standard basis {u σ } for V p (Σ) is given (see [3]) in terms of p-admissible colorings σ of the spine of a handlebody of genus g whose boundary is Σ where the set of colors is {0, 1, 2, . . ., d p − 1}, and the sum of the colors at a 3-vertex is even and less than 2d p in the case that p is twice an odd prime.
All of the above elements u σ lie in S p (Σ) when p is twice an odd prime.This follows as the quantum integers (denominators of the Jones-Wenzel idempotents) are units in O p (see Corollary 2.7.1).An admissible colored trivalent graph [3] is to be interpreted, here and elsewhere, as an O p -linear combination of links.
We say a ∼ b in O p if a/b is a unit in O p .The following proposition is an elementary fact from number theory that gives us a family of units in the ring O p .
Proposition 2.3.1 ( [23]) Suppose n has at least two distinct prime factors.Then We make use of the following lemma in giving the first basis for S p (S 1 × S 1 ) in §3 whose proof will be in §2.7.
. This is called the i-th quantum integer.
We can describe the modules S p (Σ) in terms of 'mixed graph' notation in a fixed connected 3-manifold M whose boundary is Σ.By a mixed graph, we mean a padmissibly trivalent graph whose simple closed curves may be colored ω p or an integer from the set {0, 1, . . ., p − 2} where Using the surgery axiom (S2) in [3], we can choose this fixed 3-manifold to be a handlebody whose boundary is Σ. given by a mixed graph in a fixed handlebody whose boundary is Σ with the same genus.
Proof.The first statement follows from that fact that V p satisfies the second surgery axiom.The second statement follows as every 3-manifold with boundary Σ is obtained by a sequence of 2-surgeries to a handlebody of the same boundary and the definition of S p (Σ).

2.4
The First Basis for S p (S 1 × S 1 ) In this section, we assume r is an odd prime and p = 2r.We give a standard basis for S p (S 1 × S 1 ).We need the following lemma before we state our basis.
Definition 2.4.1 Let µ i be the eigenvalue for the eigenvector e i of the twist map on the Kauffman skein module of the solid torus.It is known in [1] that Lemma 2.4.1 For i = j, we have µ i − µ j is equivalent to one of the following three cases up to a unit in O p .
Proof.Without loss of generality we can assume 0 ≤ i < j ≤ d p − 1.We have p .

Now we have three cases:
1.The hypothesis implies, is a unit by Proposition (2.3.1), as −α 2(j−i)(i+j+2) p has order divisible by two distinct primes.
2. The hypothesis implies, 3. Finally the hypothesis implies that for some k ≤ r − 1, As Proof.To prove the first part, we look at all pairs (i, j) with (j − i)(i + j + 2) ≡ 0 (mod r) which automatically will satisfy i ≡ j (mod 2).This implies that i+j +2 = r.
So we have ( r−1 2 )-pairs of such (i, j).Hence the first part follows.Now for every 0 ≤ i ≤ r − 4, there are ( r−3−i 2 )-j's such that i ≡ j (mod 2).Hence, we have 2 )-pairs of such (i, j).Hence the second part follows.
Then B p is a basis for S p (S 1 ×S 1 ).
Proof.We have Let W be the matrix which expresses B p in terms of {e 0 , e As As the determinant of W is non-zero, we conclude that B p is linearly independent.  1.
By equation (2.5) and the fact 1 + α 4 p ∼ 1 + α 4 p = 1 + α −4 p , the determinant of the form (2.2) with respect to B p is a unit.Let W denote the O p -submodule of S p (S 1 × S 1 ) generated by B p .We can conclude that the form on W is unimodular.Hence W = W and so the set B p forms a basis for S p (S 1 × S 1 ) by equation (2.3).
Remark 2.4.1 This theorem and its proof are analogous to [13,Thm. 6.1] and its proof.Proof.We expand the graph in every element in the Proposition (2.3.3) in terms of linear combinations of banded links (with some simple curves are colored ω p ). Then we replace any link component (that is not colored ω p ) by a linear combination from the set {t i (ω p )| 0 ≤ i ≤ d p − 1}.Hence the result follows by doing the required surgery on all the components of the link in every summand.

Remark 2.4.2 The above result is true if we replace p by an odd prime as a corollary
of [13,Thm. 6.1].

The Frohman Kania-Bartoszynska Ideal
We can apply the results from the previous section to compute the Frohman Kania-Bartoszynska ideal using the SU (2)-theory for a special family of 3-manifolds with torus boundary.Before we do so, we review this ideal.The importance of this ideal is in being an invariant of 3-manifolds (with boundary) and an obstruction to embedding as stated in the following propositions.In general, it is not easy to compute this ideal because we have infinitely many closed connected 3-manifolds that contains N .Following his work with Masbaum in the case p an odd prime, Gilmer observed that J p (N ) is finitely generated based on his result that S p (Σ) is finitely generated in the case p twice an odd prime as well.We give a finite set of generators for this ideal for any oriented compact 3-manifold using the SU (2)−TQFT-theory which can be obtained by the following construction.Definition 2.5.2Assume L is an ordered link of two components K, J. Let N L be the manifold obtained by doing surgery in S 3 along K in the complement of J.
Proposition 2.5.3 where M i is the 3-manifold obtained by doing surgery along the component K and the component J with framing i in S 3 .
Proof.If p is an odd prime this was proved in [12].With the help of Theorem (2.4.1), the case p twice an odd prime follows in the same way.

2.6
The Quantization Functors for p = 1, and 2 In order to understand the relation between J r and J 2r when r is an odd prime.We consider the theories associated to p = 1 and p = 2.
We begin by reviewing the quantization functor for p = 1 in detail.We start by listing the ring k 1 = Z, and the surgery element Ω 1 = ω 1 = 1 for this theory defined in [1].We also have where k is the number of components of the banded link in a closed 3-manifold M .
Then (by [3, Prop.1.1]) there exits a unique cobordism generated quantization functor (V 1 , Z 1 ) that extends this invariant.In fact, this quantization functor can be described explicitly for surfaces as follows.V 1 (Σ) is the quotient of the Z-module generated by all 3-manifolds (with banded links) with boundary Σ by the radical of the following form given by This module is isomorphic to Z with any handlebody whose boundary is Σ as a gener- Now we consider the quantization functor for p = 2.We start by introducing the ring and its ring of integers used in this theory The surgery element for this theory is √ 2, and κ 2 = ζ 8 .Therefore the invariant of a closed connected 3-manifold M , which is obtained by doing surgery on S 3 along the link L, in terms of ω 2 is given by < L(ω 2 ) >, where < > denotes the Kauffman bracket. ( From this formula, we can easily verify that Now this invariant M 2 defined in [3, §2] is involutive and extended to be multiplicative, hence (by [3, Prop.1.1]) there exits a unique cobordism generated quantization functor that extends M 2 which is denoted by (V 2 , Z 2 ).The modules V 2 (Σ) carry a Hermitian sesquilinear form defined as follows.
given by By [3, 1.5 and 6.3], V 2 (S 1 × S 1 ) is generated by two elements each of which is a solid torus where the core is colored either 0 or 1.The pairing in terms of this basis is given by Here H is the Hopf link with one of the components is colored ω 2 .Finally, Here K is the 3-chain link where the middle chain is colored ω 2 .
Hence the matrix of the form , S 1 ×S 1 in terms of this basis is given by 1 0 0 4 .
If we restrict this theory to the category of nonempty connected objects and connected cobordisms between them, then we have an integral cobordism theory as before.
This follows from the fact 2 is integral as stated in the proof of [19, Thm.given by ( , Remark 2.6.1 One could similarly define S 2 (Σ) based on the invariant 2 defined in [3, §. 1.B].In this case, the basis {1, z} for V 2 (S 1 × S 1 ) over k 2 is also a basis for S 2 (S 1 × S 1 ) over O 2 .However, this theory is not useful for us in this chapter.
Then B is a basis for S 2 (S 1 ×S 1 ), and the form is unimodular on S 2 (S 1 ×S 1 ).Moreover, the matrix of the form defined in the previous definition in terms of B is given by Proof.Let ω 2 and t(ω 2 ) stands for the elements in the Kauffman skein module of the solid torus where the core is colored ω 2 and t(ω 2 ) respectively.From the definition we know that these two elements lie in The matrix of the form ( , ) S 1 ×S 1 is given by √ 2 0 0 4 √ 2 , and since the matrix of B in terms of {1, z} is given by . Then the matrix B of the form in terms of B is given by So the form restricted on W has a unit determinant.Hence W = W .Using equation 3), we get that W is all of S 2 (S 1 × S 1 ).In conclusion, {ω 2 , t(ω 2 )} is a basis for ..i g be the boundary connected sum of g solid tori where the core of the m-th torus is colored i m = 0 or 1.Also, let This set B is an orthogonal basis for V 2 (Σ), and the pairing is described as follows: Proposition 2.6.1 The above set B forms an orthogonal basis with respect to the form , 2 given by where k = i 1 + i 2 + . . .+ i g .
Proof.By [3, 1.5 With a natural order, the matrix of the form in terms of this set is given by g B (B is defined in the proof of the previous theorem).This implies that the determinant of this form is a unit.By a similar argument as in the proof of Theorem (2.6.1), the module generated by this set is all of S 2 (Σ).
We define I 2 (M ) = √ 2 M 2 for a closed 3-manifold M where M 2 as defined in Equation (2.6).Also we define the Frohman Kania-Bartoszynska ideal J 2 just as in the previous section.Now we can compute this ideal easily for all 3-manifolds using the 2-theory by making use of above results.For example, we confirm a result of Gilmer and prove it using our basis.
with framing i, j = 0 or 1 in S 3 .As in the proof of the previous proposition, one sees ) .
If we take all possibilities, we get the required result.

Relating the r-th and 2r-th Theories
For the rest of this chapter, we assume that r is an odd prime and p = 2r.
Remark 2.7.1 The results of this section are slight variations of results of [3, §. 6] and [2, §. 2].The ring k p is not exactly the same as the ring denoted this way in [3].
The ring k p will be considered as a k 2 (or a k r )-module via the homomorphisms defined below.The following is a slight variation of the maps defined in [2, §. 2].
We need the following remark to prove that these maps are well-defined.Proof.To prove that the map i r is a well-defined ring homomorphism, we show α r 2 p is a primitive 8-th root of unity.This is true, as gcd(8r, r 2 ) = r and α p is a primitive 8r-th root of unity.Similarly for j r but we consider two cases: 1.For r ≡ 1 (mod 4), we have α 1+r 2 p is a primitive 4r-th root of unity, as gcd(8r, 1+ r 2 ) = 2 and α p is a primitive 8r-th root of unity.
2. For r ≡ −1 (mod 4), we have A 1+r 2 p is a primitive 2r-th root of unity, as gcd(4r, 1 + r 2 ) = 2 and A p is a primitive 4r-th root of unity.Proof.We know that the quantum integers [i] r for 1 ≤ i ≤ r − 1 are units in the O r see [13,Lem. 4.1(iii)] and [19,Lem. 3.1(ii)].So we conclude that [i] p are units for Given any k 2 (or k r )-module, we can define a k p -module by tensoring the original module with k p over k 2 (or k r ) respectively.We let V 2 (Σ) (or V r (Σ)) be the k p −module obtained in this way.We give a relation between V 1 , V 2 , V r , and V 2r for any surface Σ, but before that we need the following slight reformulation of [2, Thm.2.1].
Theorem 2.7.1 For any closed 3-manifold M with possibly a banded link sitting inside of it we have, Proof.Theorem (2.1) in [2] states the following: Letting M = S 3 , we get as θ 1 (S 3 ) = 1.Now multiply both sides of equation (2.9) by D −β 1 (M ) 2r , and replace it by i r (D ) in the right hand side.Then the result follows from the relation between θ and I.
We let κ n to be an element that plays the role of κ 3 in [3].We define this element as follows: , if n is twice an odd prime.
Changing the weight by one multiplies the invariant n by κ n .
Lemma 2.7.2For the above ring homomorphisms.We have We are now able to give the proof of a result used in §2.2.
Proof of Lemma 2.

j r (A
where M is a 3-manifold with banded link (but not linear combination of links) sitting inside of it.
Corollary 2.7.2The map in the previous theorem defines a k p −isomorphism between V p (Σ), and V 2 (Σ) ⊗ V r (Σ).
• If r ≥ 3, then the result follows from the fact that rank( , and the second part of the lemma.

2.8
The Second Basis for S p (S 1 × S 1 ) We give new basis for S r (S 1 × S 1 ) that will be used in constructing another basis for S p (S 1 × S 1 ).To do so, we need the following lemma.
Proof.Notice that µ i = q i 2 +2i r where q r denotes the primitive r-th root of unity given by −A r .
We used the result of the fourth part of [13, Lem.(4.1)] in the last equality up to a unit.Also in the one next to last, we used the fact that 1 + q Proof.The proof of [13,Thm. 4.1] now goes through with µ 2 i playing the role of µ i and t 2j playing the role of t j .We use the previous lemma when appropriate to obtain that B 2r is a basis.To prove that B 2r+1 is a basis, we use the fact that the twist map t is an isomorphism of the Kauffman skein module of the solid torus.

Notation: We use the notation S
We defined δ i so that the following two lemmas hold.get mapped to t 4m+1 (ω 2 ) = t(ω 2 ) for some m, as t 4 is the identity map in the 2-theory.Also they get mapped to t 2j+1 (ω r ) for some 0 ≤ j ≤ d p − 1, as t p is the identity map in the SO(3)-TQFT-theory and i is odd.The later elements form the basis B 2r+1 defined in the previous theorem.In short, the above elements get mapped to ω 2 ⊗ B 2r+1 .
Hence the image of B p under F is a basis for S 2 (S 1 ×S 1 )⊗ S r (S 1 ×S 1 ), i.e generates it as required.
We do not know if this holds for higher genus surfaces, but it is clear that S p (Σ) maps into S 2 (Σ) ⊗ S r (Σ) under the map F .Proposition 2.8.1 where M i is the 3-manifold obtained by doing surgery along the component K and the component J with framing i + δ i p in S 3 .
Finally, a good question would be: "Is there a relation between the Frohman Kania-Bartoszynska ideals in the SU (2)-and the SO(3)-TQFT-theories?"An answer is given by the following theorem.the value of colored ribbon graphs under the covariant functor F will be given in §3.3.
Finally in §3.4,we state and prove the main result.

Quantum Invariants of 3-Manifolds
Fix a strict modular category (V, {v i } i∈I ) with ground ring K and a rank D ∈ K.The material of this section is due to Turaev [22, Ch.II].

Introduction
A result due to Lickorish and Wallace asserts that every closed oriented 3-manifold can be obtained by surgery on S 3 along a framed link.

The τ -Invariant of Closed 3-Manifolds
Let M be a closed oriented 3-manifold obtained by surgery on S 3 along a framed link L. The τ -invariant of (M, Ω) associated to (V, D) where Ω is a colored ribbon graph in M is given by τ (M, Ω) = ∆ σ(L) D −σ(L)−m−1 {L, Ω}.

Some Results About Traces
We use two different notions of trace.One is the trace of a linear homomorphism (denoted by Trace) in the category of K-modules and the other one is the trace of a where δ and κ as defined before.

and this element annihilates b for all b ∈ λ 1 + λ 2 .
So the form is well-defined.As ψ is bilinear, , is bilinear.Let a be as before and b = b 1 + b 2 where b 1 ∈ λ 1 , b 2 ∈ λ 2 and b ∈ λ 3 .Since λ i = Ann(λ i )

Proposition 1 . 4 . 2
If there are even skew-cylinders C and C over Σ, and

2 (
mod 4).Here and elsewhere A p , α p are ζ 2p and ζ 4p respectively for p ≥ 3. Now, we consider the TQFT-functor (V p , Z p ) from C to the category of finitely generated projective k p -modules.The functor (V p , Z p ) is defined as follows.V p (Σ) is a quotient of the free k p -module generated by all cobordisms with boundary Σ, andZ p (M ) is the k p -linear map from V p (Σ) to V p (Σ) (where ∂M = −Σ Σ ) induced by gluing representatives of elements of V p (Σ) to M along Σ via the identification map of the first component of the boundary.If M is a closed cobordism, then Z p (M ) is the multiplication by the scalar M p defined in [3, §. 2].This invariant is normalized in two other ways.The first normalization of this invariant is I p (M ) = D p M p .Here and elsewhere M is the 3-manifold M with a reassigned weight zero, and D p = S 3 −1 p .The second normalization is

Definition 2 . 5 . 1 (
[7]) Let N be a 3-manifold with boundary, we define J p (N ) to be the ideal generated over O p by {I p (M )| where M is a closed connected 3-manifold containing N }.

Definition 2 . 6 . 1
We define S 2 (Σ) to be the O 2 -submodule of V 2 (Σ) generated by all connected vacuum states, and we define an O 2 -Hermitian sesquilinear form on S 2 (Σ)

Remark 2 . 7 . 2
If α is a primitive n-th root of unity, then α m is a primitive n gcd(n,m) -th root of unity.

Corollary 2 . 7 . 1
The quantum integers[i] p , 1 ≤ i ≤ d p are units in O p .

(3. 1 )
Here σ(L) is the signature of the linking matrix of the link L, and m is the number of components of L, and ∆ = {U − } where U − denotes the diagram for the unknot with a single double point and writhe -(L i ))F (Γ(L, λ) ∪ Ω),where col(L) is the set of all mappings from the set of components of L to I (the set of simple objects), and Γ(L, λ) is the ribbon graph obtained by coloring the i-th component of L by V λ(i) .Here F is the covariant functor defined in [22, Ch.I] which assigns to a V-colored ribbon graph in R 3 an element of the ground ring.

Lemma 3 . 5 . 1 Theorem 3 . 5 . 1 ( 3 . 6 )Corollary 3 . 5 . 1 τ
Let L be a p-periodic link, such that L * = L/Z p .Then {L} ≡ {L * } p (mod J p ). (3.4) Proof.Let us start with any coloring of L say λ, either λ is p-periodic or not.Let us assume that λ is not p-periodic, i.e Γ(L, λ) is not invariant under the rotation by 2π/p about the z-axis.Hence the i-th rotation of Γ(L, λ) ( the rotation by 2iπ/p) represents a ribbon graph with the same value under F (since F is an isotopy invariant) and different coloring denoted by λ i .So the term with a non-periodic coloring occurs p times.Hence we reduce the summation on the left-hand side to the periodic colorings.Now the result follows from corollary (3.4.1) and the fact that the periodic colorings of L are in one-to-one correspondence with the colorings of L * (by restriction).We introduce the notion κ = ∆D −1 .Now, we are ready to give a relation between the quantum invariants of M and M * .Over any modular category with integrally closed ground ring K; we haveI(M ) ≡ κ δ I(M * ) p (mod J p ),(3.5)for some integer δ.Proof.We assume that M and M * are obtained by surgery on S 3 along L and L * respectively.I(M ) = (∆D −1 ) σ(L) D −pm {L} ≡ (∆D −1 ) σ(L) D −pm {L * } p (mod J p ) by lemma(3.5.1) ≡ (∆D −1 ) σ(L)−pσ(L * ) ((∆D −1 ) σ(L * ) ) p (D −m ) p {L * } p (mod J p )≡ κ δ I(M * ) p (mod J p ).Here δ = σ(L) − pσ(L * ).(M ) ≡ κ δ D p−1 τ (M * ) p (mod J p ),(3.7) Now by Proposition (2.3.2),we know (e i , e i ) ∼ D p .Therefore the determinant of the form (2.2) with respect to the set {e 0 , e 1 , . . ., e dp−1 , and 6.3] B is a basis.The result now follows from equation (2.7), and the computations for V 2 (S 1 × S 1 ) after that equation.Let H i 1 i 2 ...i g be the boundary connected sum of g solid tori where the core of the m-th torus is colored t i m (ω 2 ) for i m = 0, or 1. {H i 1 i 2 ...ig | (i 1 , i 2 , . .., i g ) is a g-tuple over {0, 1}}.The above set B forms a basis for S 2 (Σ).Proof.Let (S 1 × S 2 ) ij denote S 1 × S 2 formed by gluing two solid tori whose cores are colored t i (ω 2 ), and t j (ω 2 ) where i, j ∈ {0, 1}.Let us look at the pairing (H i 1 i 2 ...ig , H j 1 j 2 ...jg