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HKR-type invariants of \(4\)-thickenings of \(2\)-dimensional CW complexes. (English) Zbl 1031.57019

The well-known Andrews-Curtis conjecture of combinatorial group theory asks if any finite presentation of the trivial group can be changed into the empty presentation using only “Andrews-Curtis” moves, namely inversion and permutation of generators and of relations, conjugation of relators by generators, multiplication of one generator (or relator) by another one, and the addition or removal of a generator together with a relator equal to that generator. This conjecture has strong ties to low-dimensional topology, since any finite presentation can be translated into a topological object, such as a CW-complex or handlebody, whose fundamental group has the given presentation. Although there are many potential counterexamples, often with topological origins, invariants to detect them as counterexamples have been elusive.
In this paper, the authors consider \(4\)-dimensional handlebodies obtained as thickenings of \(2\)-dimensional CW complexes, and construct invariants of these under 1- and 2-handle slides, and the birth and death of canceling 1-handle/2-handle pairs, operations which alter the presentation of the fundamental group of the handlebody by Andrews-Curtis moves. Based on the “HKR framework” of Hennings, Kauffman, and D. E. Radford [M. Hennings, J. Lond. Math. Soc. (2) 54, 594-624 (1996; Zbl 0882.57002) and L. H. Kauffmann and D. E. Radford, J. Knot Theory Ramifications 4, 131-162 (1995; Zbl 0843.57007)], the invariant uses a presentation of a \(4\)-dimensional thickening by a framed link in \(S^3\), and takes as its input data a finite dimensional unimodular ribbon Hopf algebra and an element in a quotient of its center which determines a trace function on the algebra. The authors determine a subset \({\mathcal{T}}^4\) of trace elements which yield invariants of \(4\)-thickenings under \(2\)-handle slides and 1-handle/2-handle cancellations. In \({\mathcal{T}}^4\), two subsets are identified and studied: \({\mathcal{T}}^3\subset {\mathcal{T}}^4,\) a subset of elements which factor as a product of a \(3\)-manifold invariant of the boundary and an invariant which depends only on the Euler characteristic and signature of the thickening; and \({\mathcal{T}}^2\subset {\mathcal{T}}^4,\) a subset of elements which give invariants depending on the \(2\)-dimensional spine and the second Whitney number of the thickening.

MSC:

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57M20 Two-dimensional complexes (manifolds) (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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