Michael Freedman

fl. 1951, Los Angeles, California, United States

TopologyFields Medal 1986

American topologist who proved the four-dimensional Poincaré conjecture and classified simply-connected topological 4-manifolds; awarded the Fields Medal in 1986.

Michael Hartley Freedman was born in Los Angeles in 1951 and studied at the University of California, Berkeley, before earning his PhD at Princeton in 1973 under William Browder with a dissertation on codimension-two surgery. After short positions at Berkeley and the Institute for Advanced Study he settled at the University of California, San Diego.

In a series of papers culminating in his 1982 Annals article "The topology of four-dimensional manifolds", Freedman proved the topological Poincaré conjecture in dimension four and classified simply-connected closed topological 4-manifolds by their intersection forms. The key step — showing that Casson handles are homeomorphic to standard open handles — required an infinite decomposition argument of extraordinary subtlety. Combined with Donaldson's gauge-theoretic results announced shortly afterward, his work revealed the bizarre gap between the topological and smooth categories in dimension four, including exotic smooth structures on $\mathbb{R}^4$ . He received the Fields Medal at the 1986 ICM in Berkeley and the National Medal of Science in 1987.

In 1997 Freedman moved to Microsoft, where he founded and directed Station Q in Santa Barbara, applying his expertise in low-dimensional topology to the design of a fault-tolerant topological quantum computer based on non-abelian anyons. His work there helped establish topological quantum computation as a serious approach to quantum hardware.

Workplaces: University of California, San Diego, Microsoft Station Q / Microsoft Quantum, Santa Barbara

United States

Award-winning work

Fields Medal 1986Freedman proved the Poincaré conjecture in dimension four: every closed simply-connected topological 4-manifold with the homotopy type of the 4-sphere is homeomorphic to it. To do this he showed that Casson handles — infinite towers of immersed discs that had resisted analysis for a decade — are in fact homeomorphic to genuine open 2-handles, a result obtained via a delicate infinite decomposition argument (Bing–Borsuk-style reimbedding and capped gropes). Combined with his classification of simply-connected closed topological 4-manifolds by their intersection form and Kirby–Siebenmann invariant, this completed the topological classification picture in dimension four. It also revealed that dimension four is the boundary between low- and high-dimensional topology, and it paved the way for the discovery that

\mathbb{R}^4

admits uncountably many exotic smooth structures.

Contributions, linked to the library

Great problemPoincaré conjectureProved the conjecture in the topological category in dimension four (1982); the remaining classical case, dimension three, was settled by Perelman in 2003.Great problemSmooth four-dimensional Poincaré conjectureHis results, combined with Donaldson's gauge theory, imply that dimension four behaves differently from all others and that exotic smooth structures on

\mathbb{R}^4

exist; whether the smooth four-dimensional Poincaré conjecture holds remains open.Library topicSmooth manifoldsCentral figure in the classification theory of four-dimensional manifolds.