#### Algebraic and Geometric Topology 3 (2003),
paper no. 2, pages 33-87.

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

### Ivelina Bobtcheva, Maria Grazia Messia

**Abstract**.
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 T^4 of trace elements which define invariants of
4-thickenings under 2-deformations. In T^4 two subsets are identified :
T^3, which produces invariants of 4-thickenings normalizable to
invariants of the boundary, and T^2, 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.
**Keywords**.
Hennings' invariant, Hopf algebras, CW complexes, 4--thickenings

**AMS subject classification**.
Primary: 57N13.
Secondary: 57M20, 57N10,16W30.

**DOI:** 10.2140/agt.2003.3.33

**E-print:** `arXiv:math.QA/0206307`

Submitted: 22 July 2002.
(Revised: 27 November 2002.)
Accepted: 10 January 2003.
Published: 27 January 2002.

Notes on file formats
Ivelina Bobtcheva, Maria Grazia Messia

Dipartimento di Scienze Matematiche, Universita di Ancona

Via Brece Bianche 1, 60131, Ancona, Italy

Email: bobtchev@dipmat.unian.it

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