Topologically nontrivial counterexamples to Sard's theorem I

Tuesday, December 5, 2017 - 15:00 to 15:50
Thackeray 427

Abstract or Additional Information

My talk is based on my recent joint work with Pawel Goldstein and Pekka Pankka. We prove the following dichotomy:
if n=2,3 and fC1(Sn+1,Sn) is not homotopic to a constant map,
then there is an open set ΩSn+1 such that rankDf=n on Ω and f(Ω) is dense in Sn, while
for any n4, there is a map fC1(Sn+1,Sn) that is not homotopic to a constant map and such that
rankDf<n everywhere. The proofs are based on a mixture of methods: generalized Hopf invariant, Hodge decomposition, and 
the Freudenthal suspension theorem. In the first talk I will discuss the case n=2,3 and in the second one the case n4