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
and
an element in a quotient of its center, which determines a trace function on
. We study
the subset
of trace elements which define invariants of 4–thickenings under 2–deformations. In
two subsets are
identified : ,
which produces invariants of 4–thickenings normalizable to invariants of the boundary,
and ,
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
is studied in details.
We conjecture that
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
is identified as a subalgebra of a quotient of its center.