7-dimensional closed simply-connected and spin manifolds having 2nd integral cohomology classes whose squares are not divisible by 2 and stable fold maps on them

This article presents families of 7-dimensional closed and simply-connected manifolds and fold maps on them such that squares of 2nd integral cohomology classes may not be divisible by 2. Fold maps are higher dimensional versions of Morse functions. The author has launched and been challenging the following new area: geometric and constructive studies of higher dimensional, closed and simply-connected manifolds. They are central objects in classical algebraic topology and differential topology. They were classified via algebraic and abstract objects in the last century and their understanding has been studied via concrete algebraic topological theory such as concrete bordism theory since the beginning of this century by Crowley, Kreck, Wang and so on. Fold maps are fundamental objects in the new area and the author has obtained families of these manifolds and fold maps on the manifolds. The present paper presents a new explicit result on the new area.