Investigating additional structures on preimages of regular values of smooth maps induced from the manifolds via the Reeb spaces and applications

Abstract

The Reeb space of a generic map is the space of all connected components of preimages of the map. Reeb spaces are fundamental and useful tools in the theory of Morse functions and higher dimensional variants and their applications to geometry. These spaces are polyhedra compatible with natural simplicial structures of the manifolds in considerable cases. For example, for Morse functions, fold maps, and proper stable maps, which are useful and fundamental tools in these studies, they are polyhedra whose dimensions are equal to those of the manifolds of the targets. In the 2010s, Hiratuka and Saeki found that the top-dimensional homology group of the Reeb space does not vanish for smooth maps such as proper stable maps having some preimages containing a component which is not (oriented) null-cobordant and which has no singular points. Recently the author found some extended versions of this fact formulating and using cobordism-like groups of equivalence classes of smooth and closed manifolds. The present paper concerns cases where manifolds and preimages with no singular points of maps may have additional algebraic topological or differential topological structures other than differentiable structures and orientations. We show a theorem for this case. We also give some explicit applications of the result where the preimages have spin or spin^c structures induced from the manifolds of the domains and structures of framed manifolds for example. As important applications, we present new explicit construction of spin cobordisms and spin^c cobordisms for example.