Mathematics Problem Archive

(T. Kerler) [Cyclotomic integer TQFT’s] (1) Find explicit/computable bases for the Vp(Σ g) as free modules over Z[ζp]. (2) Show that Vp can be extende...

(T. Kerler) [Homological TQFT’s] (1) Find the irreducible components and ring structure (w.r.t ⊕ and⊗) of Q∗. (2) Determine whether all strictly homol...

(T. Kerler) [Length = 1 TQFT’s] (1) Describe and construct algebraic L = 1 -extensions of Γ g -representations to TQFT’s, preferably as “simple” gener...

(T. Kerler) [ q/l -solvable and Casson TQFT’s] (1) Lift the 1/1-solvable TQFT’s of Casson type over Fp to a universal 1/1- solvable TQFT’s of Casson t...

(T. Kerler) [3-dim cobordisms from Hopf algebras] (1) Find further relations on Alg, besides the ones arising from the axiomat- ics of Hopf algebras, ...

(T. Kerler) [Extended and half-projective TQFT’s] (1) Describe in how far an ETQFT V with circle category C can differ from V C, thus introducing a equ...

(T. Kerler) [Non-semisimple vs. semisimple TQFT’s, the dou - ble conjecture] (1) Clarify the difference in the content of VC andVC! Are there homologic...

(1) Find (and classify) all semi-simple monoidal categories (w ith finitely many isomorphism classes of simple objects). (2) Find (and classify) (fini...

(Y. Kawahigashi) Suppose we have a three-dimensional TQFT. Can we determine whether it arises from a fusion rule algebra and 6j -symbols? If yes, can ...

(Y. Kawahigashi) Suppose we have two fusion rule algebras with 6j -symbols and that two TQFT’s arising from them are isomorphi c. What relation do we ...

(Y. Kawahigashi) Suppose we have a TQFT arising from a fusion rule algebra with 6j -symbols. Using a fusion rule subalgebra and 6j - symbols restricte...

(Y. Kawahigashi) Suppose we have a semisimple ribbon cat- egory C with finitely many isomorphism classes of simple objects. If the S - matrix is inver...

(Y. Kawahigashi) Suppose we have a semisimple ribbon cat- egory C1 with finitely many isomorphism classes of simple objects, bu t the S -matrix is not...

(Y. Kawahigashi) Suppose we have a semisimple ribbon cat- egory C1 with a degenerate S -matrix as in Problem 9.6. By the method in [288], we can also ...

(Y. Kawahigashi) There are some fusion rule algebras with 6j -symbols that do not seem to arise from quantum groups in [14] and more conjectured candi...

(N. Sato) Find a subfactor which can distinguish lens spaces L(7, 1) and L(7, 2). Moreover, find a subfactor to classify 3-manifolds as well as possib...

(N. Sato) Construct a well-defined state sum type invariant from a strongly amenable subfactor. Note that, unlike the Ponzano-Regge model, we do not h...

(N. Sato) Let us consider the Turaev-Viro-Ocneanu invariant from a subfactor with a degenerate braiding. Then, find a desc ription of this invariant a...

Can the Casson invariant of an integral homology 3-sphere M be characterized by the signature of a certain 4-manifold bo unded by M?...

(V. Turaev) Relate this surgery formula for the Casson-Wal- ker-Lescop invariant with that of Lescop [251]. 32The normalization here is that λ CW(M ) ...

(C. Lescop) Are the Cappell-Lee-Miller Casson-type SU (n)- invariants of finite type? If so, what are their degrees and th eir weight systems?...

(M. Polyak) Define an invariant λ of a pair (M, σ) of a closed 3-manifold M and a spin structure σ on M such that λCWL(M ) = ∑ σ λ(M, σ) for any close...

(M. Polyak) Is there a “Rokhlin invariant” of a pair (M, α) of a closed 3-manifold M and a spin c structure α on M? (See Question 10.21.)...

(M. Polyak) By presenting 3-manifolds by surgery along framed links in S3, we can regard an invariant of 3-manifolds as an invari- ant of framed links...

F as d (ZM)/F as d+1(ZM) (resp.F b d (ZM)/F b d+1(ZM)) is torsion free for each d....

A(∅; Z) is torsion free. 10.2.2 Do finite type invariants distinguish homology 3-sph eres?...

Finite type invariants distinguish integral homology 3-spheres. (See Conjecture 11.2.) 10.2.3 Dimensions of spaces of finite type invariants A finite ...

Determine the dimension of the space of primitive finite type invariants of integral homology 3-spheres of each degree d. Equivalently, deter- mine th...

Describe Vogel’s algebra Λ, say, by giving complete sets of generators and relations of Λ....

Find a constructive combinatorial presentation of each fini te type invariant of integral homology 3-spheres, and, in part icular, of the Casson invar...

(J. Roberts) What is the space of 3-manifolds?...

The map (50) is an isomorphism. 35The Yd -equivalence is also called the ( d − 1)-equivalence (due to Goussarov) in some literatures. This conjecture ...

{M ∼ Y2d S3}/ ∼ Y2d+1 is torsion free for each d....

(T. Ohtsuki) Define a product M1◦ M2 of integral homol- ogy 3-spheres M1 and M2 which is related, by (50), to the product of Jacobi diagrams given by ...

(M. Polyak, see [153, “Theorem 4”]) Let F be an oriented compact surface. Two homology cylinders C and C ′ over F are Yd -equivalent if and only if v(...

(F. Deloup) Classify the monoid (for orthogonal sum) of isomorphism classes of quadratic forms qσ....

(G. Massuyeau) Describe the quotient set {spin closed 3-manifolds}/∼ Y s d, in particular, for d = 2, 3....

(F. Deloup, G. Massuyeau) Describe the quotient set {spin c closed 3-manifolds}/∼ Y c d, in particular, for d = 2, 3....

(F. Deloup) Is there a lift of arg γ(qσ) to a mod 16 invariant? This would give a finite type invariant of degree 1 in the spin c Goussarov-Habiro the...

For each rational homology 3-sphere M, calculate Z L M O(M ) for all degrees....

The LMO invariant distinguishes integral homology 3-spheres. (See Conjecture 10.9.)...

Does there exist an integral/rational homology 3-sphere M such that Z L M O(M ) = Z L M O(S3)? 11.3 Characterization of the image of the LMO invariant...

Characterize those elements of ˆA(∅)conn which are of the form log Z L M O(M ) for integral/rational homology 3-spheres....

Construct the LMO invariant with coefficients in a finite field....

Construct the LMO invariant (or the theory of finite type invariants) in arrow diagrams. 11.5 Refinements of the LMO invariant (T. Le) As mentioned in...

(T. Le, V. Turaev) Define the LMO invariant Z L M O(M, σ) of the pair of a closed 3-manifold M and a spin structure σ of M such that Z L M O(M ) = ∑ σ...

(T. Le, V. Turaev) For every element ξ∈ H 1(M, Z) construct an extension of Z L M O(M, ξ) of the LMO invariant such that when ξ = 0 one recovers the u...

(1) Find a surgery formula for the Kuperberg-Thurston in- variant [237] in terms of the Chern-Simons series of Questio n 3.12 (2) Compare the Kuperber...

(D. Thurston) Do configuration spaces of [237] have torsion in Z-homology? Does such torsion deduce a torsion invariant of h omology 3-spheres?...

(R. Benedetti) Are torsions actually sensitive only to the (pL)-homotopy immersion classes of (pL)-knots? If one fix a C - homotopy immersion class of...