Fold maps, framed immersions and smooth structures

TL;DR

This paper develops a cohomology theory for fold maps, proves a splitting theorem for its representing spectrum, and relates cobordism groups of fold maps to framed immersions and diffeomorphism groups.

Contribution

It introduces a new cohomology theory for fold maps and establishes a splitting theorem linking cobordism groups to framed immersions and diffeomorphism groups.

Findings

Splitting theorem for the spectrum of fold maps cohomology theory

Cobordism group decompositions for even and odd q

Partial splitting of Thom spectra in the Madsen-Weiss framework

Abstract

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.