What is Poincaré Conjecture?

The Poincaré conjecture is one of the seven Millennium Prize Problems, for which the Clay Mathematics Institute offered a $1,000,000 prize for the first correct solution. It was finally solved by Grigori Perelman in 2006, for which he won the Fields Medal. The problem was first conjectured by the famous mathematician Henri Poincaré in 1904. It proved to be one of the toughest problems in all of mathematics and troubled many mathematicians for a period of more than 100 years. However, the Poincaré conjecture is the only solved Millennium problem. In this article, we'll try to understand the conjecture but will not go into its proof as it is supposed to be extremely complicated even for seasoned mathematicians. I assume knowledge of advanced set theory, advanced calculus and topology.

The Poincare conjecture states that: "Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere" or in other words, The Poincare Conjecture asks whether two the two properties of spheres - compactness and simply connectedness, are enough to characterize spheres.

Let us first try to understand these properties.

A space X is said to be compact if every open covering A of X contains a finite sub collection that also covers X. But what do we mean by open covering A?

A collection A of subsets of a space X is to said to cover X, or to be a covering of X, if the union of the elements of A is equal to X. It is called an open covering of X if its elements are open subsets of X.

Going by the definition of compact space, we can conclude that the real line R is not compact because the covering of R by open intervals  contains no finite subcollection that covers R.

I would also like to discuss the Heine–Borel theorem here. The theorem states that - For a subset S of Euclidean space Rn, the following two statements are equivalent:

1.S is closed and bounded

2.Every open cover of S has a finite subcover, that is, S is compact.

This theorem implies that a Euclidean n-sphere is compact.

Now, a manifold is simple connected if every loop in the manifold can be continuosly contracted to a point. The following figure illustrates that a 2-sphere is simply connected.

Now, as we know that a 2-sphere is compact as well as simply connected, Poincare proved that if every loop can be continuously tightened to a point, then the surface is topologically homeomorphic to a 2-sphere.

So, the Poincaré conjecture asserts that the same is true for 3-dimensional spaces. The conjecture can also be stated as:

A compact smooth n-dimensional manifold that is Homotopy Equivalent to the n-sphere Sn$S^n$ must be homeomorphic to Sn$S^n$

Perleman proved the conejcture with the help of Ricci Flow As stated above, the proof is way beyond the scope of this article as well as the scope of my mathematical knowledge!