Talk by Dr. Alex Zupan (Dept. of Mathematics, University of Nebraska)

4/25/2019  Wilson Hall 1-134  4:10-5:00pm


In its original form, the Poincaré Conjecture posits that any 3-dimensional space that is homotopy equivalent to the standard 3-dimensional sphere is actually homeomorphic to the 3-sphere.  The conjecture has since been generalized to all dimensions, in which the word "homeomorphic" can be replaced with other notions of equivalence (like diffeomorphism), with various solutions and counterexamples in different dimensions and categories.  Notably, Michael Freedman earned a Fields Medal in 1986 in part for his solution to the topological version of the Poincaré Conjecture in dimension 4.  In this talk, we focus on the Smooth 4-dimensional Poincaré Conjecture, the only version of the generalized Poincaré Conjecture yet to be solved.  We will discuss recent work showing that each member of a particular family of homotopy 4-spheres, which encompasses many interesting historical examples, is diffeomorphic to the standard 4-sphere.  This is joint work with Jeffrey Meier.