Download PDFOpen PDF in browserAn Exotic 4sphereEasyChair Preprint no. 9575, version 310 pages•Date: January 25, 2023AbstractIt has not been known whether or not there are any exotic 4spheres: such an exotic 4sphere would be a counterexample to the smooth generalized Poincare conjecture in dimension 4. Some plausible candidates are given by Gluck twists, but many cases over the years were ruled out as possible counterexamples. In the paper the resulting solution to the last generalized Poincare conjecture is presented by giving a precise construction of a discrete exotic 4sphere (Berkovich analytic spaces and the RichterGebert’s Universality theorem help). An exotic triangulation for a 4sphere (which is provided in the paper) refutes the smooth 4dimensional Poincare conjecture. Note that Pachner moves are a way to manipulate triangulations, since two triangulated manifolds are PLequivalent, there is a finite sequence of Pachner moves transforming both into another. On the other hand, every PL nsphere (any n) becomes polytopal after finitely many derived subdivisions (subset of Pachner moves/bistellar flips). Thus, two piecewise linear structures on a nsphere are unequivalent if and only if the related npolytopes are unequivalent. In other words, we come to discrete convex geometry. The idea in the paper finds common ground with padic numbers, Rado graph and beyond, since a valuation can be defined on the graph of all triangulations. Keyphrases: Berkovich analytic spaces, discrete geometry, Exotic nspheres, Exotic smooth structures, inverse limit, Pachner moves, Piecewiselinear manifolds, Poincare conjecture, Rado graph, RichterGebert’s Universality theorem, simplicial complexes, Subdivisions, triangulations, Valuation, Weighted simplicial complexes
