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

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