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

Bobtcheva, Ivelina; Messia, Maria Grazia

doi:10.2140/agt.2003.3.33

Bobtcheva, Ivelina ¹ ; Messia, Maria Grazia ¹

¹Dipartimento di Scienze Matematiche, Università di Ancona, Via Brece Bianche 1, 60131, Ancona, Italy

Algebraic and Geometric Topology, Tome 3 (2003) no. 1, pp. 33-87

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Résumé

The HKR (Hennings–Kauffman–Radford) framework is used to construct invariants of 4–thickenings of 2–dimensional CW complexes under 2–deformations (1– and 2– handle slides and creations and cancellations of 1–2 handle pairs). The input of the invariant is a finite dimensional unimodular ribbon Hopf algebra A and an element in a quotient of its center, which determines a trace function on A. We study the subset T4 of trace elements which define invariants of 4–thickenings under 2–deformations. In T4 two subsets are identified : T3 ⊂T4, which produces invariants of 4–thickenings normalizable to invariants of the boundary, and T2 ⊂T4, which produces invariants of 4–thickenings depending only on the 2–dimensional spine and the second Whitney number of the 4–thickening. The case of the quantum sl(2) is studied in details. We conjecture that sl(2) leads to four HKR–type invariants and describe the corresponding trace elements. Moreover, the fusion algebra of the semisimple quotient of the category of representations of the quantum sl(2) is identified as a subalgebra of a quotient of its center.

DOI : 10.2140/agt.2003.3.33

Keywords: Hennings' invariant, Hopf algebras, CW complexes, 4–thickenings

Affiliations des auteurs :

Bobtcheva, Ivelina ¹ ; Messia, Maria Grazia ¹

¹ Dipartimento di Scienze Matematiche, Università di Ancona, Via Brece Bianche 1, 60131, Ancona, Italy

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Bobtcheva, Ivelina; Messia, Maria Grazia. HKR-type invariants of 4–thickenings of 2–dimensional CW complexes. Algebraic and Geometric Topology, Tome 3 (2003) no. 1, pp. 33-87. doi: 10.2140/agt.2003.3.33

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