How to include fermions into general relativity by exotic smoothness

  • Torsten Asselmeyer-Maluga
  • Carl H. Brans
Research Article


The purpose of this paper is two-fold. At first we will discuss the generation of source terms in the Einstein–Hilbert action by using (topologically complicated) compact 3-manifolds. There is a large class of compact 3-manifolds with boundary such as a torus given as the complement of a (thickened) knot admitting a hyperbolic geometry, denoted as hyperbolic knot complements in the following. We will discuss the fermionic properties of this class of 3-manifolds, i.e. we are able to identify a fermion with a hyperbolic knot complement. Secondly we will construct a large class of space-times, the exotic \({\mathbb {R}}^{4}\), containing this class of 3-manifolds naturally. We begin with a topological trivial space, the \({\mathbb {R}}^{4}\), and change only the differential structure to obtain many nontrivial 3-manifolds. It is known for a long time that exotic \({\mathbb {R}}^{4}\)’s generate extra sources of gravity (Brans conjecture) but here we will analyze the structure of these source terms more carefully. Finally we will state that adding a hyperbolic knot complement will result in the appearance of a fermion as source term in the Einstein–Hilbert action.


Source terms Einstein–Hilbert action Fermions as knot complements Exotic \({\mathbb {R}}^{4}\) Adding matter by adding 3-manifolds 



This work was partly supported (T.A.) by the LASPACE grant. The authors acknowledged for all mathematical discussions with Duane Randall, Robert Gompf and Terry Lawson. Furthermore we thank the two anonymous referees for pointing out some errors and limitations in a previous version as well for all helpful remarks to increase the readability of the paper.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.German Aerospace Center (DLR)BerlinGermany
  2. 2.Loyola UniversityNew OrleansUSA

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