Reconstruction of Deligne classes and cocycles. (English) Zbl 1300.53003

Potsdam: Univ. Potsdam, Mathematisch-Naturwissenschaftliche Fakultät (Diss.). xii, 87 p. (2007).
Summary: In this thesis we mainly generalize two theorems from Mackaay-Picken and Picken (2002, 2004). In the first paper, Mackaay and Picken show that there is a bijective correspondence between Deligne 2-classes \(\xi \in \check{H}^2(M,\mathcal{D}^2)\) and holonomy maps from the second thin-homotopy group \(\pi_2^2(M)\) to \(U(1)\). In the second one, a generalization of this theorem to manifolds with boundaries is given: Picken shows that there is a bijection between Deligne 2-cocycles and a certain variant of 2-dimensional topological quantum field theories.
In this thesis we show that these two theorems hold in every dimension. We consider first the holonomy case, and by using simplicial methods we can prove that the group of smooth Deligne \(d\)-classes is isomorphic to the group of smooth holonomy maps from the \(d^{th}\) thin-homotopy group \(\pi_d^d(M)\) to \(U(1)\), if \(M\) is \((d-1)\)-connected.
We contrast this with a result of Gajer (1999). Gajer showed that Deligne \(d\)-classes can be reconstructed by a different class of holonomy maps, which not only include holonomies along spheres, but also along general \(d\)-manifolds in \(M\). This approach does not require the manifold \(M\) to be \((d-1)\)-connected. We show that in the case of flat Deligne \(d\)-classes, our result differs from Gajers, if \(M\) is not \((d-1)\)-connected, but only \((d-2)\)-connected. Stiefel manifolds do have this property, and if one applies our theorem to these and compare the result with that of Gajers theorem, it is revealed that our theorem reconstructs too many Deligne classes. This means, that our reconstruction theorem cannot live without the extra assumption on the manifold \(M\), that is our reconstruction needs less informations about the holonomy of \(d\)-manifolds in \(M\) at the price of assuming \(M\) to be \((d-1)\)-connected.
We continue to show, that also the second theorem can be generalized: By introducing the concept of Picken-type topological quantum field theory in arbitrary dimensions, we can show that every Deligne \(d\)-cocycle induces such a \(d\)-dimensional field theory with two special properties, namely thin-invariance and smoothness. We show that any \(d\)-dimensional topological quantum field theory with these two properties gives rise to a Deligne \(d\)-cocycle and verify that this construction is surjective and injective, that is both groups are isomorphic.


53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53C29 Issues of holonomy in differential geometry
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
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