Notes on explicit fold maps on 7-dimensional manifolds
Naoki Kitazawa · arXiv (Cornell University) · 2019
A fold map is a smooth map at each singular point of which it is represented as a suspension of a Morse function: in other words a product of a Morse function and the identity map on an open ball. A special generic map is a fold map such that the Morse function before is a natural height function on an unit disc. The class of special generic maps includes a Morse function with just two singular points on a closed manifold, characterizing a sphere topologically (except 4-dimensional cases) as the Reeb's theorem shows, and canonical projections of unit spheres. It has been known that exotic homotopy spheres do not admit special generic maps into Euclidean spaces whose dimensions are sufficiently high and smaller than the dimensions of the spheres. 7-dimensional exotic homotopy spheres do not admit special generic maps into 4-dimensional spaces for example. We can easily obtain special generic maps on fundamental manifolds such as ones represented as connected sums of products of two standard spheres and in considerable cases, smooth manifolds resembling topologically them and different from them do not admit special generic maps. These interesting results are owing to studies of Saeki, Sakuma, Wrazidlo, and so on, since 1990s. In the present paper, we present new results on explicit smooth maps including fold maps on 7-dimensional manifolds into the 4-dimensional Euclidean space and meanings in algebraic topology and differential topology of manifolds. Moreover, the author obtained related results before motivated by the studies before and they are reviewed in the presentation of the new results. We also present new discussions and results related to the results for 7-dimensional manifolds and maps on them for fold maps between manifolds of general dimensions.