Title:An integral Euler cycle in normally complex orbifolds and Z-valued Gromov-Witten type invariants

Abstract:We define an integral Euler cycle for a vector bundle $E$ over an effective orbifold $X$ for which $(E, X)$ is (stably) normally complex. The transversality is achieved by using Fukaya-Ono's "normally polynomial perturbations" and Brett Parker's generalization to "normally complex perturbations." Two immediate applications in symplectic topology are the definition of integer-valued genus-zero Gromov--Witten type invariants for general compact symplectic manifolds using the global Kuranishi chart constructed by Abouzaid-McLean-Smith, and an alternative proof of the cohomological splitting theorem of Abouzaid-McLean-Smith for Hamiltonian fibrations over $S^2$ with integer coefficients.