SPACES OF JETS WITHOUT COMPLICATED SINGULARITIES

摘要

Complements to discriminants of smooth mappings A discriminant is the subset in the space of functions consisting of functions (mappings) having singularities of a specified type. There are a lot of mathematical objects that can be described as the complement to a (well chosen) discriminant. As an example, one can consider the spaces of smooth mappings M~m → R~n without complicated singularities that arise naturally in the theory of pseudoisotopies. If the set of forbidden singularities has codimen-sion ≥ (m + 2) (respectively, ≥ (m + 3)) in the jet space J~k(M~m, R~n) (which means that the corresponding discriminant has codimension ≥ 2 (respectively. ≥ 3), then the set of mappings without singularities of the specified type is homology (respectively, weakly homotopy) equivalent to the space of admissible sections of the jet bundle. This statement is a manifestation of the general Smale-Hirsch principle.