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The Millennium Museum·Topology

Solved

The Poincaré Conjecture

Is every simply connected, closed three-dimensional manifold homeomorphic to the three-sphere?

Solved by Grigori Perelman in 2002–2003, the only Millennium problem solved so far, and by a person. He declined the prize.

The exhibit

The problem, three ways

Curious

Imagine a universe that is finite and has no edge. If every loop you could draw in it can be shrunk to a point, must it be, as far as its shape goes, a sphere? Henri Poincaré asked in 1904. Grigori Perelman proved the answer is yes in 2002 and 2003.

Undergraduate

A closed 3-manifold with trivial fundamental group is homeomorphic to S³. The analogues in higher dimensions were proved first: by Smale for dimension five and above, and by Freedman for dimension four. Perelman completed Hamilton's Ricci flow programme, which deforms a metric towards roundness and cuts out the singularities that form.

Specialist

Perelman's entropy functionals, non-collapsing and canonical neighbourhood theorems control Ricci flow with surgery, yielding Thurston's geometrisation conjecture, of which Poincaré is a corollary. Detailed expositions by Kleiner and Lott, Morgan and Tian, and Cao and Zhu appeared in 2006.

What would count

  • Settled. The Clay Institute awarded its prize to Perelman on 18 March 2010.

Fidelity traps

  • The Formal Conjectures file states the conjecture; the register records no formal proof of it.

Who is attacking it

No one needs to: it is solved.

The history

9 events
  1. 1900s
  2. 1904

    posed

    Poincaré's question

    Henri Poincaré asks whether a closed three-dimensional manifold whose loops can all be contracted must be the three-sphere.

    Source

  3. 1960s
  4. 1961

    progress

    Dimension five and above

    Stephen Smale proves the analogue in every dimension from five up.

    Source

  5. 1980s
  6. 1982

    progress

    Dimension four

    Michael Freedman proves the topological analogue in dimension four.

    Source

  7. 1982

    progress

    Ricci flow

    Richard Hamilton introduces the Ricci flow, the tool that would eventually settle the conjecture.

    Source

  8. 2000s
  9. 2000-05-24

    prize

    A Millennium Prize Problem

    The Clay Mathematics Institute names the conjecture one of seven carrying a $1,000,000 prize.

    Source

  10. 2002-11-11

    progress

    Perelman's first preprint

    Grigori Perelman posts the first of three preprints on arXiv, completing Hamilton's programme.

    Source

  11. 2003

    progress

    The second and third preprints

    Perelman posts the remaining two preprints, on Ricci flow with surgery and on finite extinction time.

    Source

  12. 2006

    prize

    A Fields Medal, declined

    Perelman is awarded the Fields Medal and declines it, as detailed expositions of his proof appear.

    Source

  13. 2010s
  14. 2010-03-18

    prize

    The prize awarded, and declined

    The Clay Institute awards the Millennium Prize to Perelman, who declines it.

    Source

Follow and discuss

All discussion

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