What is the biggest unsolved problem in topology

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In summary, the biggest unsolved problem in topology is the Poincaré conjecture for differentiable 4-manifolds, which remains open despite being solved for 3-manifolds by Perelman in 2003. Friedman did show it for topological 4-manifolds in 1982, but the case of differentiable 4-manifolds is still unsolved.
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kurt.physics
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What is the biggest unsolved problem in topology
 
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The Poincaré conjecture for differentiable 4-manifolds? (As opposed to the Poincaré conjecture for 3-manifolds, which Perelman solved.)
 
  • #3
CRGreathouse said:
The Poincaré conjecture for differentiable 4-manifolds? (As opposed to the Poincaré conjecture for 3-manifolds, which Perelman solved.)

Could you explain the Poincaré for 4-manifolds? Is it harder to prove than 3-manifolds?
 
  • #4
wasnt that proved by freedman in around 1980?
 
  • #5
mathwonk said:
wasnt that proved by freedman in around 1980?

Friedman showed it for topological 4-manifolds in 1982, but the case of differentiable 4-manifolds is still open as far as I know.

And I really don't know much about topology, I was just throwing out an answer.
 

1. What is topology?

Topology is a branch of mathematics that studies the properties of space and the relationships between objects within that space. It focuses on the study of continuous transformations and how they affect the shape and structure of objects.

2. What is the biggest unsolved problem in topology?

The biggest unsolved problem in topology is the Poincaré conjecture, which was first proposed by the French mathematician Henri Poincaré in 1904. It states that any closed 3-dimensional manifold (a type of geometrical object) is topologically equivalent to a 3-dimensional sphere.

3. How has the Poincaré conjecture impacted the field of topology?

The Poincaré conjecture has been a driving force for research in topology for over a century. It has led to the development of new techniques and theories, as well as the discovery of other important conjectures and theorems. It has also sparked collaborations between mathematicians from different subfields, leading to advancements in related areas of mathematics.

4. What progress has been made towards solving the Poincaré conjecture?

In 2002, Russian mathematician Grigori Perelman presented a proof of the Poincaré conjecture, using a combination of techniques from topology and geometry. However, his proof has not yet been fully accepted by the mathematical community as some details are still being verified. If accepted, it would be considered one of the most significant achievements in mathematics in recent years.

5. What implications would a solution to the Poincaré conjecture have?

A solution to the Poincaré conjecture would have far-reaching implications in the field of topology and mathematics as a whole. It would provide a deeper understanding of the structure of 3-dimensional space and the behavior of continuous transformations. It would also have applications in other areas such as physics, computer science, and engineering.

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