For a compact \(G_2\) manifold \((X^7,\varphi)\), \(\varphi\in\Omega^3(X)\), the author constructs an abelian group \(\mathrm{CDiv}(X,\varphi)=\{\sum_{i=1}^kq_iN_i|q_i\in\mathbb Z,N_i \text{ being coassociative submanifolds of }X\}\), an analogue of the group of divisors in a projective complex manifold, and an abelian group \(\mathrm{MPic}(X,\varphi)=\{\text{monopole gerbes}\}\), an analogue of the Picard group of a projective complex manifold. A monopole gerbe is a gerbe with connection \((\mathcal G, F)\) over \(X\) such that (1) there is an open cover \(\{U_i\}_{i\in I}\) of \(X\) equipped with local trivializations of \(\mathcal G\), and a connection \(2\)-form \(F_i\) satisfying \(\star(F_i\wedge \star\varphi)=d\phi_i\) for some \(\phi_i\in C^\infty(U_i)\), (2) the curvature of \(F\) is the harmonic representative of \(c_1(\mathcal G)\). Here the star operator and harmonic forms are defined by using the metric \(g\) on \(X\) induced from \(\varphi\).
N. Hitchin [AMS/IP Stud. Adv. Math. 23, 151–182 (2001; Zbl 1079.14522)] defined a canonical way to associated a gerbe with a connection to a codimension-3 submanifold. When \((X,\varphi)\) is irreducible, Hitchin’s construction defines a group homomorphism \(m:\mathrm{CDiv}(X,\varphi)\rightarrow \mathrm{MPic}(X,\varphi)\). By taking the first Chern class, there is also a surjective group homomorphism \(\mathrm{MPic}(X,\varphi)\rightarrow H^3(X,\mathbb{Z})\). On the other hand, the author also considers the topological restrictions of coassociative representatives of classes in \(H^3(X,\mathbb Z)\). It is proved that if \((X,\varphi)\) is a compact \(G_2\)-manifold, and \(\alpha\in H^3(X,\mathbb Z)\) is represented by a coassociative submanifold \(N\subset X\), i.e., \(PD(N)=\alpha\), then \(p_1(X)\cup \alpha=6\tau-2\chi\), where \(\tau\) and \(\chi\) denote the signature and Euler characteristic of \(N\) respectively. In particular, if \(b_3(X)=1\), then \((X, \varphi)\) has no coassociative tori.