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Session 26. Physics and Differential Topology
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Exotic smoothness, foliations and torsion on 4-manifolds |
Torsten Asselmeyer-Maluga, German Aerospace Center, Germany
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The geometry of open exotic 4-manifolds like exotic \(\mathbb{R}^{4}\)
or \(S^{3}\times\mathbb{R}\) is quite mysterious but important from
the point of physics. In this talk, the relation of exotic smoothness
structures to foliations is discussed. The central invariant in this
context is the Godbillon-Vey invariant which will be related to the
parameter of the DeMichaelis-Freedman radial family of small exotic
\(\mathbb{R}^{4}\). The leaf space of these foliations is a factor
\(I\! I\! I\) von Neumann algebra in the sense of Connes. The origin
of this algebra can be understood by analyzing wildly embedded submanifolds
like the topologically embedded 3-sphere in the exotic \(S^{3}\times\mathbb{R}\).
From the geometrical point of view, these foliations are related to
hyperbolic geometry (the non-compact counterpart of non-positive scalar
curvature results for compact exotic 4-manifolds). Finally, the appearance
of torsion in exotic 4-manifolds is also discussed.
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