HKR-type invariants of 4–thickenings of 2–dimensional CW complexes

Bobtcheva, Ivelina; Messia, M. G.

doi:10.2140/agt.2003.3.33

Cited by 7 publications

(10 citation statements)

References 27 publications

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“…Finally, we observe that the HKR-type invariants of 2-complexes, defined in [4] from unimodular triangular Hopf algebras, in many cases are also…”

Section: 3mentioning

confidence: 88%

“…The present work offers another criterion concerning the class of invariants of 2-complexes which are reductions of invariants of 4-thickenings of such complexes. Surprisingly it is easy to see that many of the known invariants of 2-complexes are such reductions: all (except maybe one) numerically generated examples of Q Z/p , but also many of the HKR-type invariants constructed in [4] from triangular non-cosemisimple algebras.…”

Section: 2mentioning

confidence: 99%

“…Let L be a BOK-link. Then Proposition 5.5 in [4] states that if P L and P ′ are 2-equivalent, there exist 2-deformationsσ 0 and ξ such that the diagram…”

Section: 6mentioning

confidence: 99%

“…Now the statement would follow from the one with ξ = id. For ξ = id it has been proved in [4,Lemma 2.16] by showing that if the corresponding undotted components of L and L ′ have the same framings modulo 2, then through regular isotopy and move (ii) both diagrams can be deformed into the same standard form as a closure, through the dotted components, of a chosen braid. If the framings differ modulo 2, we need move (iii) to adjust them.…”

Section: 6mentioning

confidence: 99%

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The reduction of quantum invariants of 4-thickenings

Bobtcheva

Quinn

2005

Fund. Math.

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Abstract. We study the sensibility of an invariant of 2-dimensional CW complexes in the case when it comes as a reduction (through a change of ring) of a modular invariant of 4-dimensional thickenings of such complexes: it is shown that if the Euler characteristic of the 2-complex is greater than or equal to 1, its invariant depends only on homology. To see what is happening when the Euler characteristic is smaller than 1, we use ideas of Kerler and construct, from any tortile category, an invariant of 4-thickenings which, if the category is modular, is a normalization of the Reshetikhin-Turaev invariant, and if the category is symmetric, depends only on the spine and the second Whitney number of the thickening. Then we show that the so(3) quantum invariant at a 5th root of unity has its reduction, and this reduction is able to distinguish complexes with the same homology groups when the Euler characteristic is smaller than 1.

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“…Finally, we observe that the HKR-type invariants of 2-complexes, defined in [4] from unimodular triangular Hopf algebras, in many cases are also…”

Section: 3mentioning

confidence: 88%

Section: 2mentioning

confidence: 99%

“…Let L be a BOK-link. Then Proposition 5.5 in [4] states that if P L and P ′ are 2-equivalent, there exist 2-deformationsσ 0 and ξ such that the diagram…”

Section: 6mentioning

confidence: 99%

Section: 6mentioning

confidence: 99%

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The reduction of quantum invariants of 4-thickenings

Bobtcheva

Quinn

2005

Fund. Math.

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“…The solution of Problem B ′ associates to any braided ribbon Hopf algebra an invariant of 4-dimensional 2handlebodies under 2-deformations and implies that, if the conjecture is true, there should exist a braided ribbon Hopf algebra whose invariant distinguishes diffeomorphic but not 2-equivalent handlebodies. The search for such Hopf algebras is a non-trivial challenge, since they have to combine the properties of being unimodular, not self-dual and not semisimple (see the discussion in [11] about the the HKR-type invariants associated to an ordinary ribbon Hopf algebra). d) By restricting the map ✰ 2 1 • Ψ 2 to double branched covers of B 4 , i.e.…”

Section: Introductionmentioning

confidence: 99%

On 4-Dimensional 2-Handlebodies and 3-Manifolds

Bobtcheva

Piergallini

2012

J. Knot Theory Ramifications

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We show that for any n ≥ 4 there exists an equivalence functor [Formula: see text] from the category [Formula: see text] of n-fold connected simple coverings of B3 × [0, 1] branched over ribbon surface tangles up to certain local ribbon moves, and the cobordism category [Formula: see text] of orientable relative 4-dimensional 2-handlebody cobordisms up to 2-deformations. As a consequence, we obtain an equivalence theorem for simple coverings of S3 branched over links, which provides a complete solution to the long-standing Fox–Montesinos covering moves problem. This last result generalizes to coverings of any degree results by the second author and Apostolakis, concerning respectively the case of degree 3 and 4. We also provide an extension of the equivalence theorem to possibly non-simple coverings of S3 branched over embedded graphs. Then, we factor the functor above as [Formula: see text], where [Formula: see text] is an equivalence functor to a universal braided category [Formula: see text] freely generated by a Hopf algebra object H. In this way, we get a complete algebraic description of the category [Formula: see text]. From this we derive an analogous description of the category [Formula: see text] of 2-framed relative 3-dimensional cobordisms, which resolves a problem posed by Kerler.

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Kerler–Lyubashenko Functors on 4-Dimensional 2-Handlebodies

Beliakova

Renzi

2023

International Mathematics Research Notices

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We construct a braided monoidal functor $J_4$ from Bobtcheva and Piergallini’s category $4\textrm {HB}$ of connected 4D 2-handlebodies (up to 2-deformations) to an arbitrary unimodular ribbon category $\mathscr {C}$, which is not required to be semisimple. The main example of target category is provided by $\operatorname {\mathnormal {H}-mod}$, the category of left modules over a unimodular ribbon Hopf algebra $H$. The source category $4\textrm {HB}$ is generated, as a braided monoidal category, by a $4$-modular Hopf algebra object, and this is sent by the Kerler–Lyubashenko functor $J_4$ to the end $\int _{X \in \mathscr {C}} X \otimes X^*$ in $\mathscr {C}$, which is given by the adjoint representation in the case of $\operatorname {\mathnormal {H}-mod}$. When $\mathscr {C}$ is factorizable, we show that the construction only depends on the boundary and signature of handlebodies and thus projects to a functor $J_3^{\sigma }$ defined on Kerler’s category $3\textrm {Cob}^{\sigma }$ of connected framed 3D cobordisms. When $H^*$ is not semisimple and $H$ is not factorizable, our functor $J_4$ has the potential of detecting diffeomorphisms that are not 2-deformations.

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HKR-type invariants of 4–thickenings of 2–dimensional CW complexes

Cited by 7 publications

References 27 publications

The reduction of quantum invariants of 4-thickenings

The reduction of quantum invariants of 4-thickenings

On 4-Dimensional 2-Handlebodies and 3-Manifolds

Kerler–Lyubashenko Functors on 4-Dimensional 2-Handlebodies

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