This is a joint work with Masamichi Takase (Seikei University, Japan).
Given a singular map f : Mn ! Rp of a closed manifold of dimension n
with n  p  1, we consider the following problem: for a standard projection
 : Rm ! Rp with m > n  p  1, determine if there exists an immersion or
embedding  : Mn ! Rm such that f =   .
Rm

Mn
f / ? 9: non-singular
oo7
Rp
Such a map  can be considered as a “resolution of singularities” of f.
For generic maps of surfaces into the plane, Haefliger (1960) obtained a
necessary and sufficient condition for such a map to be lifted to an immersion
into R3. Burlet-Haab (1985) showed that any Morse function on a surface
can be lifted to an immersion into R3. Other than these, some results are
known, but most of them concern maps between equidimensional manifolds.
In this talk, we consider special generic maps f : Mn ! Rp that have
only definite fold as their singularities. For various dimension pairs (n, p),
we give answers to the existence problem of immersion or embedding lifts
into Rn+1. In particular, for the cases where p = 1 and 2 we obtain complete
results. Our techniques are related to Smale–Hirsch theory of immersions,
topology of the space of immersions, relation between the space of topological
immersions and that of smooth immersions, sphere eversions, differentiable
structures of homotopy spheres, diffeomorphism group of spheres, free group
actions on the sphere, etc.