By Osamu Saeki

The quantity develops a radical concept of singular fibers of time-honored differentiable maps. this can be the 1st paintings that establishes the foundational framework of the worldwide learn of singular differentiable maps of unfavourable codimension from the point of view of differential topology. The ebook comprises not just a normal thought, but additionally a few particular examples including a few very concrete functions.

This is a truly fascinating topic in differential topology, because it indicates a gorgeous interaction among the standard idea of singularities of differentiable maps and the geometric topology of manifolds.

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**Topology of Singular Fibers of Differentiable Maps**

The quantity develops a radical idea of singular fibers of regularly occurring differentiable maps. this can be the 1st paintings that establishes the foundational framework of the worldwide research of singular differentiable maps of destructive codimension from the perspective of differential topology. The e-book includes not just a common thought, but in addition a few particular examples including a couple of very concrete purposes.

**A First Course in Algebraic Topology**

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6. 6 Examples of Stable Maps of 4-Manifolds In this chapter, we give explicit examples of C ∞ stable maps of 4-manifolds into R3 . e. special generic maps (see [51, 43, 44, 48, 49]). Such maps have singular ﬁbers of types I0 , II00 , and III000 , and have no other singular ﬁbers. Furthermore, the source 4-manifolds of such maps always have even Euler characteristics. Here we construct more complicated maps having a singular ﬁber of type III8 such that the source 4-manifold has odd Euler characteristic.

Deformation of functions on the 2-disk R π Fig. 9. 8). Therefore, we obtain a C ∞ stable map g : M → R3 by gluing f |f −1 (R3 Int Y ) , g1 and g2 , where gY = g1 ∪ g2 (see Fig. 7 again). Note that the singular set S(g) is the union of a 2-sphere component consisting of deﬁnite fold points and a projective plane component containing the set of cusp points. 1. The smooth closed 4-manifold M is diﬀeomorphic to CP 2 or CP 2 . 56 6 Examples of Stable Maps of 4-Manifolds Proof. Let π : R3 → R be a projection.

Furthermore, this codimension does not depend on a particular choice of f ∈ Tpr (n, p). Proof. 1, F(f ) is a union of strata. 2 (2). By virtue of the above lemma, the codimension of F makes sense, and we denote it by κ(F). For an equivalence class F of ﬁbers with respect to with κ = κ(F), let ∂ F be the set of equivalence classes G of ﬁbers with respect to of codimension 7 Generalities 63 κ + 1 such that G(f ) ⊂ F(f ) F(f ) for every f ∈ Tpr (n, p). For G ∈ ∂ F, we take a proper Thom map f ∈ Tpr (n, p) with G(f ) = ∅.