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Chapitre D'ouvrage Année : 2023

Refined Bobtcheva–Messia invariants of 4-dimensional 2-handlebodies

Résumé

In this paper we refine our recently constructed invariants of 4-dimensional 2-handlebodies up to 2-deformations. More precisely, we define invariants of pairs of the form (W,ω), where W is a 4-dimensional 2-handlebody, ω is a relative cohomology class in H2(W,∂W;G), and G is an abelian group. The algebraic input required for this construction is a unimodular ribbon Hopf G-coalgebra. We study these refined invariants for the restricted quantum group U=Uqsl2 at a root of unity q of even order, and for its braided extension U~=U~qsl2, which fits in this framework for G=Z/2Z, and we relate them to our original invariant. We deduce decomposition formulas for the original invariants in terms of the refined ones, generalizing splittings of the Witten-Reshetikhin-Turaev invariants with respect to spin structures and cohomology classes. Moreover, we identify our non-refined invariant associated with the small quantum group U¯=U¯qsl2 at a root of unity q whose order is divisible by 4 with the refined one associated with the restricted quantum group U for the trivial cohomology class ω=0.

Dates et versions

hal-04230788 , version 1 (06-10-2023)

Identifiants

Citer

Anna Beliakova, Marco de Renzi. Refined Bobtcheva–Messia invariants of 4-dimensional 2-handlebodies. Essays in Geometry, 1, EMS Press, pp.387-432, 2023, IRMA Lectures in Mathematics and Theoretical Physics, ⟨10.4171/IRMA/34/20⟩. ⟨hal-04230788⟩
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