TITLE: Triangulations of exotic 4-spheres
QUESTION [8 upvotes]: Are there explicit examples of triangulations of exotic 4-spheres?

REPLY [2 votes]: The current status of the smooth Poincare conjecture in dimension 4 is presented in the paper:
Michael Freedman, Robert Gompf, Scott Morrison and Kevin Walker "Man and machine thinking about the smooth 4-dimensional Poincare conjecture" in Quantum Topology, Volume 1, Issue 2 (2010), pp. 171–208 (arXiv)
Thus the Cappel-Shaneson approach seem to fail by Akbuluts work. Now there is only possible construction via the Gluck twist with a real 2-knot (i.e. knotted 2-sphere), i.e. a knot not coming from a 3-dimensional (classical) knot.