An Exotic 4-sphere

EasyChair Preprint 9575, version 3

Abstract

It has not been known whether or not there are any exotic 4-spheres: such an exotic 4-sphere 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 4-sphere (Berkovich analytic spaces and the Richter-Gebert’s Universality theorem help).

An exotic triangulation for a 4-sphere (which is provided in the paper) refutes the smooth 4-dimensional Poincare conjecture. Note that Pachner moves are a way to manipulate triangulations, since two triangulated manifolds are PL-equivalent, there is a finite sequence of Pachner moves transforming both into another. On the other hand, every PL n-sphere (any n) becomes polytopal after finitely many derived subdivisions (subset of Pachner moves/bistellar flips). Thus, two piecewise linear structures on a n-sphere are unequivalent if and only if the related n-polytopes are unequivalent. In other words, we come to discrete convex geometry.

The idea in the paper finds common ground with p-adic numbers, Rado graph and beyond, since a valuation can be defined on the graph of all triangulations.

Keyphrases: Berkovich analytic spaces, Exotic n-spheres, Exotic smooth structures, Pachner moves, Piecewise-linear manifolds, Poincare conjecture, Rado graph, Richter-Gebert’s Universality theorem, Subdivisions, Valuation, Weighted simplicial complexes, discrete geometry, inverse limit, simplicial complexes, triangulations

@booklet{EasyChair:9575,
  author    = {Valerii Sopin},
  title     = {An Exotic 4-sphere},
  howpublished = {EasyChair Preprint 9575},
  year      = {EasyChair, 2023}}