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We define an integral Euler cycle for a vector bundle $E$ over an effectiveorbifold $X$ for which $(E, X)$ is (stably) normally complex. Thetransversality is achieved by using Fukaya-Ono's "normally polynomialperturbations" and Brett Parker's generalization to "normally complexperturbations." Two immediate applications in symplectic topology are thedefinition of integer-valued genus-zero Gromov--Witten type invariants forgeneral compact symplectic manifolds using the global Kuranishi chartconstructed by Abouzaid-McLean-Smith, and an alternative proof of thecohomological splitting theorem of Abouzaid-McLean-Smith for Hamiltonianfibrations over $S^2$ with integer coefficients.

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