An Exotic 4-sphere

EasyChair Preprint no. 9575, version 4

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.

This paper is a survey of recent advances towards the last generalized Poincare conjecture (connections with Berkovich analytic spaces, the Richter-Gebert’s Universality theorem and cluster algebras are presented) with some approaches to the resulting solution.

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

@Booklet{EasyChair:9575,
  author = {Valerii Sopin},
  title = {An Exotic 4-sphere},
  howpublished = {EasyChair Preprint no. 9575},

  year = {EasyChair, 2023}}