An Invariant for Transverse Coassociative 4-Folds

TL;DR

This paper introduces a $Z_2$-valued invariant for transverse coassociative 4-folds with spin structures, serving as an obstruction to their deformation and linking to connected sums and near-symplectic geometry.

Contribution

It defines a new invariant for transverse coassociative 4-folds, establishes its role as an obstruction, and relates it to connected sums and parity functions in geometry.

Findings

Invariant obstructs deformation between coassociatives

Connected sum determined by the invariant

Invariant computed for specific $Sp(1)$-invariant examples

Abstract

We define a $\mathbb{Z}_2$-valued invariant for transversely-intersecting coassociative $4$-folds equipped with spin structures. Our main result shows this invariant provides an obstruction to separating two such coassociatives through a family of transverse coassociative deformations. We further prove that there is a canonical generalized connected sum of two such transverse coassociatives whose diffeomorphism type is determined by our invariant. When one coassociative is a graph over the other, we relate our invariant to the parity function in near-symplectic geometry. Finally, we discuss conjectural consequences for non-compactness phenomena and compute our invariant for the $Sp(1)$-invariant coassociatives discovered by Harvey and Lawson.