1966 Fields Medal (4)
Reason for Award
Worked indifferential topologywhere he proved the generalizedPoincaré conjecturein dimension n≥5: Every closed, n-dimensional manifoldhomotopy-equivalentto then-dimensional sphereishomeomorphicto it. Introduced the method ofhandle-bodiesto solve this and related problems.
Laureates
Stephen Smale,
United States of America
Other works in the same year
- Did joint work withHirzebruchinK-theory; proved jointly withSingertheindex theoremof elliptic operators oncomplex manifolds; worked in collaboration withBottto prove afixed point theoremrelated to the "Lefschetz formula".
- Used technique called "forcing" to prove the independence inset theoryof theaxiom of choiceand of thegeneralized continuum hypothesis. The latter problem was the first ofHilbert's problemsof the 1900 Congress.
- Built on work ofWeilandZariskiand effected fundamental advances inalgebraic geometry. He introduced the idea ofK-theory(theGrothendieck groupsand rings). Revolutionizedhomological algebrain his celebrated "Tohokupaper"