A Khovanov type invariant derived from an unoriented HQFT for links in thickened surfaces (Q2863007)

Homotopy quantum field theories (HQFTs) were introduced by Turaev in 1999. An (oriented) (\(n+1\)) HQFT with target a space, \(X\), assigned a module to each (oriented) \(n\)-manifold, endowed with a structure map to \(X\) (called an \(X\)-manifold, some details omitted here). It assigns a module homomorphism to an \(X\)-cobordism. When \(n=1\) and \(X\) is a \(K(\pi,1)\), such an HQFT corresponds to a crossed \(\pi\)-algebra. In [Topology Appl. 159, No. 3, 833--849 (2012; Zbl 1244.57059)], the author showed how an unoriented version of a \((1+1)\)-HQFT could be defined, and explored the corresponding algebra.NEWLINENEWLINEFor each oriented link in \(S^3\), Khovanov defined a graded chain complex whose graded Euler characteristic was the Jones polynomial of the link. This construction was related, by Bar-Natan in 2005, to an oriented TQFT, and then \textit{V. Turaev} and \textit{P. Turner} [Algebr. Geom. Topol. 6, 1069--1093 (2006; Zbl 1134.57004)] constructed a link homology using an unoriented TQFT. In this paper, for an oriented link diagram, \(D\) in an oriented compact surface, \(F\), the author constructs a link homology by using an unoriented HQFT with target \(X = K(H_1(F;\mathbf{F_2}),1)\). The resulting link homology yields an invariant of links in the oriented \(I\)-bundle of a compact surface, \(I\) being the unit interval. The paper ends with some example computations.