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

Global classification of isolated singularities in dimensions (4,3) and (8,5)

“…Moreover, under the finiteness hypothesis, ϕ(M m , N n ) = 1 if and only if M m is the connected sum of a smooth fibration over N n with an exotic sphere and not a fibration itself. A short outline of this and related results for Lefschetz fibrations can be found in [2]. These results comfort the idea that, in general, ϕ is infinite.…”

“…The simplest exceptional case is that of pairs of surfaces, which admits an elementary treatment and is completely understood (see [2] for explicit computations). In [22] the authors described a series of examples in dimensions (4,3), (8,5) and (16,9).…”

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

“…In [1] the authors found that, in small codimension 0 ≤ m − n − 1 ≤ 3, if ϕ(M m , N n+1 ) is finite, then ϕ(M m , N n+1 ) ∈ {0, 1}, except for the exceptional pairs of dimensions (m, n + 1) ∈ {(2, 2), (4,3), (4,2), (5,2), (6,3), (8,5)}. Notice that (5,3) was inadvertently included in [1] among the exceptional pairs.…”

“…The simplest exceptional case is that of (pairs of) surfaces, which is completely understood by elementary means (see [2] for explicit computations). Very little is known for the other exceptional and generic (i.e.…”

The size of some vanishing and critical sets

“…Note that the other extreme is well represented in the recent years, as quite some effort oriented towards the maps with finite critical sets has been done, not only for one dimensional, but also for higher dimensional target manifolds [1,2,3,8,9,10].…”