DETECTING EXOTIC SPHERES VIA FOLD MAPS (Local and global study of singularity theory of differentiable maps)

Abstract

In this survey article, we present two subgroup filtrations of the group of homotopy spheres whose definitions are both based on the existence of certain fold maps subject to index constraints. Both filtrations have recently been introduced and studied by the author in order to obtain new insights into global singularity theory of fold maps from high dimensional manifolds into Euclidean spaces. We discuss fundamental relations of our filtrations to other known filtrations of geometric topology. Moreover, we show how our results can be applied to compute an invariant of Saeki for the Milnor 7-sphere, as well as the value of Banagl's TFT-type aggregate invariant on certain exotic spheres including Kervaire spheres. Along the way, we raise some problems for future study.