Fold maps, framed immersions and smooth structures

Abstract:

For each integer $q\ge 0$, there is a cohomology theory $\mathbf {A}_1$ such that the zero cohomology group $\mathbf {A}_1^0(N)$ 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.