Arnold S. Shapiro
Arnold Samuel Shapiro (1921 in Boston, Massachusetts – 1962 in Newton, Massachusetts) was an American mathematician known for his eversion of the sphere. He also was the author of an article on Clifford algebras and periodicity with Raoul Bott, later redone by Michael Atiyah and Bott.
In 1949 Shapiro was a student of Norman Steenrod at University of Michigan. He wrote an article "Group extensions of compact groups" and was awarded a master’s degree.
In 1950 Shapiro was a student of André Weil at University of Chicago. With a dissertation "Cohomology relations in fiber bundles", he was awarded a Ph.D.
He continued his studies at the Institute for Advanced Studies from 1955 to 57.Raoul Bott was also at the Institute at that time; he recounted his mathematical contacts in an AMS-MAA invited address August 9, 1988, in Providence Rhode Island:
In 2000 Allyn Jackson interviewed Bott, who then revealed Shapiro’s part in the Periodicity Theorem. He explained that there was a controversy in dimension 10 about the homotopy of the unitary group.
In 1957 Shapiro published an extension of Dehn's lemma after a method of Papakyriakopoulos. In 1960 Shapiro contributed to the Bourbaki Seminar his "Algèbres de Clifford et periodicité des groupes πK(BO))". The topic was taken up again in 1964 as Clifford modules by Bott and Atiyah with Shapiro named as an author, though he had died.
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