Free immersions and panelled web 4-manifolds

Abstract

We show that if a compact, oriented 4-manifold admits a coassociative([Formula: see text])-free immersion into [Formula: see text], then its Euler characteristic [Formula: see text] and signature [Formula: see text] vanish. Moreover, in the spin case, the Gauss map is contractible, so that the immersed manifold is parallelizable. The proof makes use of homotopy theory, in particular, obstruction theory. As a further application, we prove a non-existence result for some infinite families of 4-manifolds that have not been addressed previously. We give concrete examples of parallelizable 4-manifolds with complicated non-simply-connected topology.