An Invariant for Transverse Coassociative 4-Folds

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.