What are some big unsolved problems in topology?
Topology is a branch of mathematics that studies the properties of geometric objects that are continuous. It has applications in many fields, including physics, engineering, and computer science.
Some of the big unsolved problems in topology include:
- The Poincaré conjecture: This conjecture states that every simply connected, closed 3-manifold is homeomorphic to a 3-sphere. It was first proposed by Henri Poincaré in 1904, and it was finally proved by Grigori Perelman in 2003.
- The sphere theorem: This theorem states that every compact, simply connected 4-manifold is homeomorphic to a 4-sphere. It was first proposed by Stephen Smale in 1961, and it remains unsolved.
- The smooth Poincaré conjecture: This conjecture states that every simply connected, closed 4-manifold is diffeomorphic to a 4-sphere. It was first proposed by Michael Freedman in 1982, and it remains unsolved.
- The knot complement problem: This problem asks whether every knot complement is homeomorphic to a 3-sphere. It was first proposed by John Milnor in 1961, and it remains unsolved.
- The homology sphere problem: This problem asks whether every homology sphere is homeomorphic to a sphere. It was first proposed by Edwin Spanier in 1940, and it remains unsolved.
Related Questions:
- What is the Poincaré conjecture? Every simply connected, closed 3-manifold is homeomorphic to a 3-sphere.
- What is the sphere theorem? Every compact, simply connected 4-manifold is homeomorphic to a 4-sphere.
- What is the smooth Poincaré conjecture? Every simply connected, closed 4-manifold is diffeomorphic to a 4-sphere.
- What is the knot complement problem? Every knot complement is homeomorphic to a 3-sphere.
- What is the homology sphere problem? Every homology sphere is homeomorphic to a sphere.
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