  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.
