Reeb spaces of 1st derivatives of proper submersions of certain classes

Abstract

We study the (canonical) 1st derivatives of {\it proper} submersions represented as height functions and belonging to a certain class: a proper map means a map the preimage of a compact set by which is always compact. We investigate their {\it Reeb spaces}. They are the quotient spaces defined by the equivalence relations on the manifolds of the domains where we identify two points in a same connected component of a same level set of them. They have been important since the establishment of theory of Morse functions, in the 20th century, They are in certain tame situations $0$- or $1$-dimensional and graphs naturally. Related facts have been shown by Gelbukh and Saeki in the 2020s for certain proper smooth real-valued functions. In non-proper cases, even related explicit theory has been difficult, except some previously given case of the author. Our study is on a new related case.