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

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 ${\mathcal{T}}^4$ of trace elements which define invariants of 4–thickenings under 2–deformations. In ${\mathcal{T}}^4$ two subsets are identified : ${\mathcal{T}}^3\subset {\mathcal{T}}^4$, which produces invariants of 4–thickenings normalizable to invariants of the boundary, and ${\mathcal{T}}^2\subset {\mathcal{T}}^4$, 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 $sl\left(2\right)$ is studied in details. We conjecture that $sl\left(2\right)$ 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 $sl\left(2\right)$ is identified as a subalgebra of a quotient of its center.