  For each integer q>0 there is a cohomology theory such that the zero
cohomology group of a manifold N of dimension n is a certain group of cobordism
classes of proper fold maps of manifolds of dimension n+q into N. We prove a
splitting theorem for the spectrum representing the cohomology theory of fold
maps. For even q, the splitting theorem implies that the cobordism group of
fold maps to a manifold N is a sum of q/2 cobordism groups of framed immersions
to N and a group related to diffeomorphism groups of manifolds of dimension
q+1. Similarly, in the case of odd q, the cobordism group of fold maps splits
off (q-1)/2 cobordism groups of framed immersions.
  The proof of the splitting theorem gives a partial splitting of the homotopy
cofiber sequence of Thom spectra in the Madsen-Weiss approach to diffeomorphism
groups of manifolds.
