Abstract
In this chapter we describe topological and numerical characteristics of singular sets of smooth maps, such as cohomology classes dual to sets of critical points and critical values; the invariants of maps defined by these classes; their connections with standard topological characteristics of the source and target manifolds; the structure of spaces of smooth maps without singularities of some given type; restrictions on the number and coexistence of singular points.
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Notes
In view of smoothing theorems, M and N, and together with them Jk(M, N), can be assumed to be analytic manifolds.
Henceforth, all the results discussed in this section admit a natural “complexification”: the assertions remain valid if real manifolds and bundles are replaced by complex ones, the homology with coefficients in ℤ2 by that with coefficients in ℤ, the Stiefel-Whitney classes by the Chern classes, and so on.
Of course, here all the terms with even coefficients can be ignored, but the point is that these formulas remain valid after “complexification” (see the footnote on page 187).
Here “sufficiently general” means that the jet extensions of the map are assumed to be non-degenerate, and hence, in contrast to the real case, is not the same as “almost any”: for example, in general a compact manifold does not admit “sufficiently general” maps into ℂn. An analogy with the real case arises here only in the case of Stein manifolds.
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© 1998 Springer-Verlag Berlin Heidelberg
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Arnold, V.I., Goryunov, V.V., Lyashko, O.V., Vasil’ev, V.A. (1998). The Global Theory of Singularities. In: Singularity Theory I. Encylopedia of Mathematical Sciences, vol 6. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58009-3_4
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DOI: https://doi.org/10.1007/978-3-642-58009-3_4
Publisher Name: Springer, Berlin, Heidelberg
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