An Invariant for Transverse Coassociative 4-Folds

TL;DR

The paper defines a $ obreakspace\mathbb{Z}_2$-valued invariant $\mu$ for transversely intersecting coassociative 4-folds in a $G_2$-manifold, using spin structures and the $G_2$-induced relation between normal bundles along the intersection. It proves that $\mu$ is invariant under transverse coassociative deformations and ambient diffeomorphisms preserving the $G_2$-structure, and shows that any locally removable intersection component forces $\mu(\Gamma)=-1$, yielding an obstruction to separability. The work also provides a canonical generalized connected sum $X_+\#_Z X_-$ whose diffeomorphism type is determined by $\mu$, and connects the invariant to the parity in near-symplectic geometry in graphical settings. The Harvey–Lawson $Sp(1)$-invariant coassociatives are used as explicit test cases, and conjectural implications for non-compactness phenomena and moduli-space boundaries are discussed, with connections to a forthcoming framework (G25) relating $\mu$ to Lawlor-neck gluing and coassociative blow-down.

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.

Figures (6)

• Figure 1: Constructing $S^4\#_{S^1}S^4$ via a handle attachment. The dashed lines represent discs $D^2_\pm$ bounding the $S^1\subset S^4_\pm$, which have been pushed into the respective $5$-balls. The generalized connected sum $S^4\#_{S^1}S^4$ is the boundary of this picture, hence the boundary of a $D^3$ bundle over the $S^2$ formed by the two dashed lines.
• Figure 2: The fibres $\Sigma^\pm$ with the intersection circles of $M^\pm_{\epsilon,c}$ and $H_{s\epsilon}$ depicted. The intersection circle bounds a disc in the fibre $\Sigma^+$.
• Figure 3: Coassociative translations of $H_{s\epsilon}$ so that its intersection circle with $M^+_{\epsilon,c}$, contained in the fibre $\Sigma^+$, shrinks and disappears.
• Figure 4: The two connected components of $I_0(3)$, interchanged by the $\mathbb{Z}_2$-action $(V,V')\mapsto (V',V)$. The vertex corresponds to stabilizer $\textup{SO}(3)$, the edges to $\textup{O}(2)$, and the open regions to $K_4$. The lines $\ell_\pi$ and $\ell_{2\pi}$ are labeled on $I_0(3)$ with arrows. The diagonal $\Delta\subset \textup{SLG}(3)^2$ is also marked.
• Figure 5: The shaded triangle corresponds to $\partial_{\{\tau=0\}}\mathcal{P}\subset\bar{\mathcal{P}}$. Degenerations in one-parameter families are suggested by dotted arrows.

Theorems & Definitions (44)

• Theorem 1.1
• Theorem 1.2
• Definition 1
• Definition 2
• Lemma 1
• proof
• Definition 3
• Proposition 1
• proof
• Theorem 2.1