Fold maps, framed immersions and smooth structures

Abstract

Abstract. For each integer q ≥ 0, there is a cohomology theory A 1 such that the zero cohomology group A 0 1 (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.

Compositions of equi-dimensional fold maps

“…The fold bordism groups have been studied by many authors (see e.g. [3][4][5][6][7][8]14]). In particular, Ando [3,5,6] has proven that SFold(n, 0) is isomorphic to the stable homotopy group π S n of spheres.…”