Examples of smooth maps with finitely many critical points in dimensions $(4,3)$, $(8,5)$ and $(16,9)$

Abstract

Abstract. We consider manifolds M 2n which admit smooth maps into a connected sum of S 1 × S n with only finitely many critical points, for n ∈ {2, 4, 8}, and compute the minimal number of critical points.

On the minimal number of critical points of smooth maps between closed manifolds

“…For instance, there are topological obstructions preventing singular points to cluster together. Specifically, the authors of [15] proved the following:…”

“…Thus the proof is complete for codimension at most 2. In [16] we will analyse wild codimension three singularities.…”

Manifolds Which Admit Maps with Finitely Many Critical Points Into Spheres of Small Dimensions

“…Further, if m − n = 3 and there exists a smooth function M m → N n with finitely many critical points, all of them cone-like, then ϕ(M m , N n ) ∈ {0, 1} except for the exceptional pairs of dimensions (m, n) ∈ {(5, 2), (6,3), (8,5)}. On the other hand in [11] the authors provided many nontrivial examples and showed that ϕ(M m , S n ) can take arbitrarily large even values for m = 2n − 2, n ∈ {3, 5, 9}; these examples were classified in [10] for n ∈ {3, 5}.…”

“…It would be interesting to know how accurate are our estimates -compare with the lower bounds for ϕ(M 2n−2 , S n ) obtained in [11] -in order to characterize the set of values taken by ϕ(M m , S n ).…”

Isomorphic homotopy groups of certain regular sets and their images

“…However, by using them one can produce maps with higher dimensional source and target spaces and finite nonempty A( f ) sets. For some examples of MontgomerySamelson fibrations, we referee to[7].Let M, N be compact manifolds, f : M → N be a Montgomery-Samelson fibration with finitely many singularities and y ∈ N be a regular value of f . Consider a diffeomorphism ϕ of N which fixes every point in a disk D with spherical boundary around y and moves each singular value of f to a regular value of f .…”