Free immersions and panelled web 4-manifolds

TL;DR

This paper proves that certain 4-manifolds cannot be immersed coassociatively in Euclidean 7-space unless they have trivial Euler characteristic and signature, and explores implications for their topology and parallelizability.

Contribution

It establishes new non-existence results for coassociative-free immersions of 4-manifolds using homotopy and obstruction theory, and provides examples of complex parallelizable 4-manifolds.

Findings

Euler characteristic and signature vanish for coassociative-free immersions

In the spin case, the Gauss map is contractible, implying parallelizability

Non-existence results for specific families of 4-manifolds

Abstract

We show that if a compact, oriented 4-manifold admits a coassociative-free immersion into the Euclidean 7-space then its Euler characteristic and signature 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 can not be addressed previously. We give concrete examples of parallelizable 4-manifolds with complicated non-simply-connected topology.