Free immersions and panelled web 4-manifolds

We show that if a compact, oriented 4-manifold admits a coassociative(*phi(0))-free immersion into R-7 , then its Euler characteristic chi(M) and signature tau(M) 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.