Mathematics Problem Archive

(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 C1 with a degenerate S -matrix as in Problem 9.6. By the method in [288], we can also ...

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

(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 ) ...

(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 ...

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...

(R. Benedetti) For every W, for every (pL)-class α1 as above, f ∗ 1 is an isomorphism. This means, in particular, that finite typ e invari- ants of Le...

(H.R. Morton) From a knot diagram find an explicit such homomorphism to some permutation group or establish that th e knot is trivial. Refinements. (1...

(Y. Nakanishi, T. Harikae [220, Conjecture 1.59 (6)]) Any link can be related to a trivial link by a sequence of (2,2)-mo ves....

Find a new proof of the existence of a universal Vassiliev in- variant of knots, presenting them by KTG’s and their operati ons....

(D. Bar-Natan, D. Thurston) For each compact Lie group G, level k, and every KTG K: Γ → R3, there exists a collection of measures µ K on the space of ...

Construct an invariant of KTG’s from configuration space in- tegrals in a natural way. Turaev [388] introduced a presentation of 3-manifolds as S1 -bu...

(D. Thurston) The shadow number of a 3-manifold is quasi-linear in its Gromov norm. That is, there exist consta nts c1 and c2 such that c1||M||≤ (shad...

(D. Thurston) Find a condition on shadow diagrams which is satisfied by shadow diagrams from alternating knots; and g ives a lower bound on the hyperb...

Construct a universal Reshetikhin-Turaev invariant and a universal Turaev-Viro-Ocneanu invariant of closed 3-mani folds, in terms of the KTG algebra....