Topology of Singular Fibers of Differentiable Maps

Cobordism Group of Morse Functions on Unoriented Surfaces

“…Let f : M → R be a Morse function on a closed surface M. One can see (see, for example, [5,8]) that the Reeb function of f in a neighborhood of the q f -image of a critical point is equivalent to one of the functions as depicted in Figure 1. In case (a), the corresponding critical point of f is a local minimum or a local maximum; in case (b) the corresponding critical point of f has index 1 with sign +1 or −1, and a sign is associated to each vertex of degree three as in Figure 1.…”

Fold maps on 4-manifolds

“…For details, see [27,29,31,33]. Note that Corollary 4.10 (1) follows also from the example constructed in [30] mentioned below. According to [24], examples as in Corollary 4.10 (2) do not exist for general dimensions.…”

“…For explicit examples of fold maps, see [26,30]. For example, an explicit example of a fold map CP 2 2CP 2 → R 3 is constructed in [30], where This can be proved by using the fact that N is parallelizable and that every continuous map of a closed surface into N is homotopic to an immersion [41].…”

Stable maps and branched shadows of 3-manifolds

“…See Section 1.2 for details. For an S-map of (M, L) into R 2 , the singular fibers over nonsimple crossings are divided into two types, types II 2 and II 3 , according to their shapes as shown in Figure 1, see Saeki [37]. We denote by II 2 (f ) and II 3 (f ) the sets of singular fibers of types II 2 and II 3 , respectively, of an S-map f : (M, L) → R 2 .…”

“…We remark that, in Saeki's book, these notations are defined for singular fibers of stable maps of orientable 4-manifolds into 3-manifolds. However, the classification of singular fibers of stable maps of orientable 3-manifolds into a plane coincide with that of singular fibers of codimension 0, 1, 2 of stable maps of orientable 4-manifolds into 3-manifolds as mentioned in [37,Remark 3.14]. For this reason, we consistently use the symbols in [37, Figure 3.4] also for the remaining types of singular fibers described below.…”