  In view of the Segal construction each category with a coherent operation
gives rise to a cohomology theory. Similarly each open stable differential
relation $R$ imposed on smooth maps of manifolds determines cohomology theories
$k^*$ and $h^*$; the cohomology theory $k^*$ describes invariants of solutions
of $R$, while $h^*$ describes invariants of so-called stable formal solutions
of $R$. We prove the bordism version of the h-principle: The cohomology
theories $k^*$ and $h^*$ are equivalent for a fairly arbitrary open stable
differential relation $R$. Furthermore, we determine the homotopy type of
$h^*$. Thus, we show that for a fairly arbitrary open stable differential
relation $R$, the machinery of stable homotopy theory can be applied to perform
explicit computations and determine invariants of solutions.
  In the case of the differential relation whose solutions are all maps, our
construction amounts to the Pontrjagin-Thom construction. In the case of the
covering differential relation our result is equivalent to the
Barratt-Priddy-Quillen theorem asserting that the direct limit of classifying
spaces $B\Sigma_n$ of permutation groups $\Sigma_n$ of finite sets of n
elements is homology equivalent to each path component of the infinite loop
space $\Omega^{\infty}S^{\infty}$. In the case of the submersion differential
relation imposed on maps of dimension $d=2$ the cohomology theories $k^*$ and
$h^*$ are not equivalent. Nevertheless, our methods still apply and can be used
to recover the Madsen-Weiss theorem (the Mumford Conjecture).

Geometric Topology;Algebraic Topology;Category Theory