By Osamu Saeki
The quantity develops an intensive thought of singular fibers of familiar differentiable maps. this can be the 1st paintings that establishes the foundational framework of the worldwide examine of singular differentiable maps of unfavorable codimension from the perspective of differential topology. The e-book comprises not just a common thought, but in addition a few particular examples including a couple of very concrete functions.
This is a truly attention-grabbing topic in differential topology, because it exhibits a gorgeous interaction among the standard concept of singularities of differentiable maps and the geometric topology of manifolds.
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Additional resources for Topology of Singular Fibers of Differentiable Maps
Example text
For example, the C 0 equivalence is clearly admissible in the above sense. We denote the C 0 equivalence relation among the fibers of elements of Tpr (n, p) by 0n,p . 2. 3. For every equivalence class F with respect to an admissible equivalence relation , and for every proper Thom map f : M → N in Tpr (n, p), the subspace F(f ) of N is a union of strata of N and is a C 0 submanifold of N of constant codimension if it is nonempty. Furthermore, this codimension does not depend on a particular choice of f ∈ Tpr (n, p).
1) If two fibers are C 0 equivalent, then they are also equivalent with respect to . (2) For any two proper Thom maps fi : Mi → Ni in Tpr (n, p) and for any points yi ∈ Ni , i = 0, 1, such that the fibers over yi are equivalent to each other with respect to , there exist neighborhoods Ui of yi in Ni , i = 0, 1, and a homeomorphism ϕ : U0 → U1 such that ϕ(y0 ) = y1 and ϕ(U0 ∩ F(f0 )) = U1 ∩ F(f1 ) for every equivalence class F of fibers with respect to , where F(fi ) is the set of points in Ni over which lies a fiber of fi of type F.
Hence, the fibers over y0 and y1 are equivalent. 30 3 Classification of Singular Fibers Fig. 8. Degeneration of fibers around the fiber of type IIIe The same argument works when the fibers over y0 and y1 are of type IIIc . When the fibers over y0 and y1 are of type I1 , we can imitate the above argument for the case of I0 ; however, we cannot take Vi to be a connected component of (fi )−1 (Ui ), since the relevant singular points are indefinite fold points. 1. 1 (i) so that the diffeomorphism ϕ0 : (f0 )−1 (y0 ) ∩ V0 → (f1 )−1 (y1 ) ∩ V1 extends to one between the whole fibers (f0 )−1 (y0 ) and (f1 )−1 (y1 ).