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Novikov's 1970 paper Pontrjagin classes, the fundamental group and some problems of stable algebra is available online. A proof of Novikov additivity was first published on pages 587-589 of the Atiyah-Singer paper "The index of elliptic operators: III." (Annals of Maths. 87, 546-604 (1968)). 1968)) mentioned in the question. There is a footnote on page 587 "We are indebted to Hirzebruch for drawing our attention to Novikov's result."

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Novikov's 1970 paper Pontrjagin classes, the fundamental group and some problems of stable algebra is available online. A proof of Novikov additivity was first published on pages 587-589 of the Atiyah-Singer paper "The index of elliptic operators: III." (Annals of Maths. 87, 546-604 (1968)).