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Algebraic Criteria for Stable Diffeomorphism of Spin $4$-Manifolds Mathew Dowling and Part 2 uses theThe Memoirs of the AMS is devoted to the publication of new research in all areas of pure and applied mathematics. The Memoirs is designed particularly to publish long papers of groups of cognate papers in book form, and is under the supervision of the Editorial Committee of the AMS journal Transactions of the American Mathematical Society. All papers are peer reviewed.