In terms of category theory, the
Gromov homotopy principle for a set valued
functor asserts
that the functor
can be induced from a homotopy functor. Similarly, we say
that the
bordism principle for an abelian group valued functor
holds if the
functor
can be induced from a (co)homology functor.
We examine the bordism principle in the case of functors given by (co)bordism
groups of maps with prescribed singularities. Our main result implies that if a
family of
prescribed singularity types satisfies certain mild conditions, then there exists an infinite loop space
such that for each smooth
manifold
the cobordism
group of maps into
with only
–singularities
is isomorphic to the group of homotopy classes of maps
. The
spaces
are relatively simple, which makes explicit computations possible even in the case
where the dimension of the source manifold is bigger than the dimension of the target
manifold.
Keywords
differential relation, h-principle, generalized cohomology
theory, singularity of a smooth map, jet, fold map, Morin
map, Thom–Boardman singularity
