This project addresses a fundamental theoretical issue in the formulation of chiral gauge theories on the lattice. The objective is to show that it is possible to maintain gauge-invariance at finite lattice spacing in the overlap formulation of chiral gauge theories on the lattice when the usual (continuum) gauge anomaly cancellation conditions are satisfied. The proposed approach for showing this focuses on global topological aspects of the problem, and has its starting point in the applicants earlier work on topological obstructions to gauge-invariance in the lattice chiral gauge theory. The plan is to extend the classical continuum limit results obtained there to finite (i.e. non-zero) lattice spacing, and then combine this with mathematical results on the topological structure of determinant line bundles to establish the desired result.