A new transversality condition on orbifolds and integer-valued Gromov-Witten type invariants

Following a proposal of Fukaya-Ono and the exploration by B. Parker, we introduce a new transversality condition, the FOP transversality condition, for sections of orbifold vector bundles $\mathcal{E} \rightarrow \mathcal{U}$ when both $\mathcal{E}$ and $\mathcal{U}$ have "normal complex structures." This notion allows one to define various integral virtual cycles on moduli spaces of pseudoholomorphic curves. Two immediate applications in symplectic topology are the definition of integer-valued Gromov-Witten type invariants in all genera for general compact symplectic manifolds using the global Kuranishi chart constructed by Abouzaid-McLean-Smith and Hirschi-Swaminathan, and an alternative proof of the cohomological splitting theorem for Hamiltonian fibrations over $S^2$ with integer coefficients by Abouzaid-McLean-Smith.